According to the Bible, we can actually “know” when we are the children of God  

Believing, obeying and loving God, have lots to do with it.

1 John. 2: 3 And hereby we do know that we know him, IF we keep his commandments.4 He that saith, I know him, and keepeth not his commandments, is a liar, and the truth is not in him.5 But whoso keepeth his word, in him verily is the love of God perfected: hereby know we that we are in him.

1 John 3:10 In this the children of God are manifest, and the children of the devil: whosoever DOETH NOT RIGHTEOUSNESS is not of God, neither he that loveth not his brother.

1 John 3:18 My little children, let us not love in word, neither in tongue; but in DEED and in truth.19 And hereby we know that we are of the truth, and shall assure our hearts before him.—24 And he that keepeth his commandments dwelleth in him, and he in him. And hereby we know that he abideth in us, by the Spirit which he hath given us.

1 John 5:2 By this we know that we love the children of God, when we love God, and keep his commandments.3 For this is the love of God, that we keep his commandments: and his commandments are not grievous.—18 We know that whosoever is born of God sinneth not; but he that is begotten of God keepeth himself, and that wicked one toucheth him not.19 And we know that we are of God, and the whole world lieth in wickedness.20 And we know that the Son of God is come, and hath given us an understanding, that we may know him that is true, and we are in him that is true, even in his Son Jesus Christ. This is the true God, and eternal life.

Mark. 12: 30 And thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy mind, and with all thy strength: this is the first commandment. 31 And the second is like, namely  this, Thou shalt love thy neighbour as thyself. There is none other commandment greater than these. 

The word “know” above, and in the verses below, comes from the Greek word ginóskó (Strong’s 1097), and it means to know, especially through personal experience. It’s for example used in Luke 1:34 when it comes to the virgin Mary and the issue of “knowing” someone really close and intimately – which she said she had not done, resulting in her amazement that she would soon be getting a son. It can also mean to just be aware of something. Naturally Jesus had knowledge of sin (he knew the meaning of it), but he never engaged  in sin, and therefore he knew no sin.

2 Cor. 5:21 For he hath made him to be sin for us, who knew no sin; that we might be made the righteousness of God in him.

Jesus knew all men, because he knew what was in man’s heart. You can’t fool God.

John 2:23 Now when he was in Jerusalem at the passover, in the feast day, many believed in his name, when they saw the miracles which he did.24 But Jesus did not commit himself unto them, because he knew all men, 25 And needed not that any should testify of man: for he knew what was in man.

It’s possible  to know God enough to realize his power and to know who he is, but in order to  really know God – as in having a close relationship with him – something more is needed. In order to be his children we also need to  glorify him, be thankful for what he has done, walk in his light, know him as he truly is, worship him, retain him in our knowledge, etc.

Romans 1:21 Because that, when they knew God, they glorified him not as God, neither were thankful; but became vain in their imaginations, and their foolish heart was darkened.22 Professing themselves to be wise, they became fools,—25 Who changed the truth of God into a lie, and worshipped and served the creature more than the Creator, who is blessed for ever. Amen.—28 And even as they did not like to retain God in their knowledge, God gave them over to a reprobate mind, to do those things which are not convenient;

There is a difference between knowing about someone, and knowing someone intimately. We have all been given the light, and we are all able to get know God. The world did not know God intimately, but if we love God and believe in him (and show it by our actions) then we are known by him. If we are not acknowledged by God, we are in big trouble.

John 1:9 That was the true Light, which lighteth every man that cometh into the world.10 He was in the world, and the world was made by him, and the world knew him not.

1 Cor. 1:21 For after that in the wisdom of God the world by wisdom knew not God, it pleased God by the foolishness of preaching to save them that believe.

1 Cor 2:12 Now we have received, not the spirit of the world, but the spirit which is of God; that we might know the things that are freely given to us of God.

1 Cor. 8:3 But if any man love God, the same is known of him.

John 16:3 And these things will they do unto you, because they have not known the Father, nor me.

John 17:3 And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent.—6 I have manifested thy name unto the men which thou gavest me out of the world: thine they were, and thou gavest them me; and they have kept thy word.7 Now they have known that all things whatsoever thou hast given me are of thee.8 For I have given unto them the words which thou gavest me; and they have received them, and have known surely that I came out from thee, and they have believed that thou didst send me.—21 That they all may be one; as thou, Father, art in me, and I in thee, that they also may be one in us: that the world may believe that thou hast sent me.22 And the glory which thou gavest me I have given them; that they may be one, even as we are one:23 I in them, and thou in me, that they may be made perfect in one; and that the world may know that thou hast sent me, and hast loved them, as thou hast loved me.24 Father, I will that they also, whom thou hast given me, be with me where I am; that they may behold my glory, which thou hast given me: for thou lovedst me before the foundation of the world.25 O righteous Father, the world hath not known thee: but I have known thee, and these have known that thou hast sent me.

It’s something positive for us to call on God, to prophesy, to cast out devils and do miracles, but if we mix this with not doing the Father’s will (i. e. being disobedient), then we are not known by God. We are not among his children, because his children are those who “doeth the will of my Father”.

Matt. 7:21 Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven.22 Many will say to me in that day, Lord, Lord, have we not prophesied in thy name? and in thy name have cast out devils? and in thy name done many wonderful works?23 And then will I profess unto them, I never KNEW you: depart from me, ye that work iniquity.

A person can’t be known by God unless he keeps his commandments, according to what we have learned, and yet it’s possible for such a known person to return to a spiritual place where he is in bondage (like the ceremonial laws).

Gal. 4:9 But now, after that ye have known God, or rather are known of God, how turn ye again to the weak and beggarly elements, whereunto ye desire again to be in bondage?10 Ye observe days, and months, and times, and years.

Jesus knows his sheep, and the sheep know him, so salvation is based on synergism. Jesus sheep can be identified through their hearing of his voice and through their following of him, which is something they do by free will.

John 10: 14 I am the good shepherd, and know my sheep, and am known of mine.15 As the Father knoweth me, even so know I the Father: and I lay down my life for the sheep.16 And other sheep I have, which are not of this fold: them also I must bring, and they shall hear my voice; and there shall be one fold, and one shepherd.—27 My sheep hear my voice, and I know them, and they follow me:—38 But if I do, though ye believe not me, believe the works: that ye may know, and believe, that the Father is in me, and I in him.

2 Tim. 2:19 Nevertheless the foundation of God standeth sure, having this seal, The Lord knoweth them that are his. And, let every one that nameth the name of Christ depart from iniquity.—20 But in a great house there are not only vessels of gold and of silver, but also of wood and of earth; and some to honour, and some to dishonour.21 If a man therefore purge himself from these, he shall be a vessel unto honour, sanctified, and meet for the master’s use, and prepared unto every good work.22 Flee also youthful lusts: but follow righteousness, faith, charity, peace, with them that call on the Lord out of a pure heart.

“He that hath my commandments, and keepeth them, he it is that loveth me: and he that loveth me shall be loved of my Father”

Joh. 14:12 Verily, verily, I say unto you, He that believeth on me, the works that I do shall he do also; and greater works than these shall he do; because I go unto my Father.—20 At that day ye shall know that I am in my Father, and ye in me, and I in you.21 He that hath my commandments, and keepeth them, he it is that loveth me: and he that loveth me shall be loved of my Father, and I will love him, and will manifest myself to him.22 Judas saith unto him, not Iscariot, Lord, how is it that thou wilt manifest thyself unto us, and not unto the world?23 Jesus answered and said unto him, If a man love me, he will keep my words: and my Father will love him, and we will come unto him, and make our abode with him.24 He that loveth me not keepeth not my sayings: and the word which ye hear is not mine, but the Father’s which sent me.

We can know the tree by its fruit, and it’s up to us how we choose to make our tree. The children of God are those who do good – show good fruit.

Matt. 12:33 Either make the tree good, and his fruit good; or else make the tree corrupt, and his fruit corrupt: for the tree is known by his fruit.34 O generation of vipers, how can ye, being evil, speak good things? for out of the abundance of the heart the mouth speaketh.

Luke 6:43 For a good tree bringeth not forth corrupt fruit; neither doth a corrupt tree bring forth good fruit.44 For every tree is known by his own fruit. For of thorns men do not gather figs, nor of a bramble bush gather they grapes.

Peter truly believed Jesus was the son of God. He was sure of it (he knew it).

John 6:68 Then Simon Peter answered him, Lord, to whom shall we go? thou hast the words of eternal life.69 And we believe and are sure that thou art that Christ, the Son of the living God.

“If you CONTINUE in my word, then are ye my disciples.” ” If ye were Abraham’s children, ye would do the works of Abraham.”

John 8:31 Then said Jesus to those Jews which believed on him, If ye continue in my word, then are ye my disciples indeed;32 And ye shall know the truth, and the truth shall make you free.—34 Jesus answered them, Verily, verily, I say unto you, Whosoever committeth sin is the servant of sin.—39 They answered and said unto him, Abraham is our father. Jesus saith unto them, If ye were Abraham’s children, ye would do the works of Abraham.—42 Jesus said unto them, If God were your Father, ye would love me: for I proceeded forth and came from God; neither came I of myself, but he sent me.— 51 Verily, verily, I say unto you, If a man keep my saying, he shall never see death.—55 Yet ye have not known him; but I know him: and if I should say, I know him not, I shall be a liar like unto you: but I know him, and keep his saying. 

We are all expected to know God’s ways.

Hebr 3:8 Harden not your hearts, as in the provocation, in the day of temptation in the wilderness: 9 When your fathers tempted me, proved me, and saw my works forty years.10 Wherefore I was grieved with that generation, and said, They do alway err in their heart; and they have not known my ways.11 So I sware in my wrath, They shall not enter into my rest.)12 Take heed, brethren, lest there be in any of you an evil heart of unbelief, in departing from the living God.13 But exhort one another daily, while it is called To day; lest any of you be hardened through the deceitfulness of sin.14 For we are made partakers of Christ, if we hold the beginning of our confidence stedfast unto the end;15 While it is said, To day if ye will hear his voice, harden not your hearts, as in the provocation.

To know God is also to accept that Jesus is come in the flesh. If Jesus came in the flesh, he did not always in the flesh, and that is also what the Bible says. Jesus has always been together with his Father in the trinity from eternity past. Yet again, we can see that we love God if we show it by our actions.

1 John 4:2 Hereby know ye the Spirit of God: Every spirit that confesseth that Jesus Christ is come in the flesh is of God:—6 We are of God: he that knoweth God heareth us; he that is not of God heareth not us. Hereby know we the spirit of truth, and the spirit of error.7 Beloved, let us love one another: for love is of God; and every one that loveth is born of God, and knoweth God.8 He that loveth not knoweth not God; for God is love.—11 Beloved, if God so loved us, we ought also to love one another.12 No man hath seen God at any time. If we love one another, God dwelleth in us, and his love is perfected in us.13 Hereby know we that we dwell in him, and he in us, because he hath given us of his Spirit.14 And we have seen and do testify that the Father sent the Son to be the Saviour of the world.15 Whosoever shall confess that Jesus is the Son of God, God dwelleth in him, and he in God.16 And we have known and believed the love that God hath to us. God is love; and he that dwelleth in love dwelleth in God, and God in him.—20 If a man say, I love God, and hateth his brother, he is a liar: for he that loveth not his brother whom he hath seen, how can he love God whom he hath not seen?21 And this commandment have we from him, That he who loveth God love his brother also.

And finally, here is 1 John. 2-3 again, with a broader context. 

1 John 2:1 My little children, these things write I unto you, that ye sin not. And if any man sin, we have an advocate with the Father, Jesus Christ the righteous:2 And he is the propitiation for our sins: and not for ours only, but also for the sins of the whole world.3 And hereby we do know that we know him, if we keep his commandments.4 He that saith, I know him, and keepeth not his commandments, is a liar, and the truth is not in him.5 But whoso keepeth his word, in him verily is the love of God perfected: hereby know we that we are in him.6 He that saith he abideth in him ought himself also so to walk, even as he walked.— 13 I write unto you, fathers, because ye have known him that is from the beginning. I write unto you, young men, because ye have overcome the wicked one. I write unto you, little children, because ye have known the Father.14 I have written unto you, fathers, because ye have known him that is from the beginning. I have written unto you, young men, because ye are strong, and the word of God abideth in you, and ye have overcome the wicked one.—17 And the world passeth away, and the lust thereof: but he that doeth the will of God abideth for ever.18 Little children, it is the last time: and as ye have heard that antichrist shall come, even now are there many antichrists; whereby we know that it is the last time. —24 Let that therefore abide in you, which ye have heard from the beginning. If that which ye have heard from the beginning shall remain in you, ye also shall continue in the Son, and in the Father. —28 And now, little children, abide in him; that, when he shall appear, we may have confidence, and not be ashamed before him at his coming.29 If ye know that he is righteous, ye know that every one that doeth righteousness is born of him.

1 John 3:1 Behold, what manner of love the Father hath bestowed upon us, that we should be called the sons of God: therefore the world knoweth us not, because it knew him not.2 Beloved, now are we the sons of God, and it doth not yet appear what we shall be: but we know that, when he shall appear, we shall be like him; for we shall see him as he is.—6 Whosoever abideth in him sinneth not: whosoever sinneth hath not seen him, neither known him.7 Little children, let no man deceive you: he that doeth righteousness is righteous, even as he is righteous.8 He that committeth sin is of the devil; for the devil sinneth from the beginning. For this purpose the Son of God was manifested, that he might destroy the works of the devil.9 Whosoever is born of God doth not commit sin; for his seed remaineth in him: and he cannot sin, because he is born of God.10 In this the children of God are manifest, and the children of the devil: whosoever doeth not righteousness is not of God, neither he that loveth not his brother.—14 We know that we have passed from death unto life, because we love the brethren. He that loveth not his brother abideth in death.15 Whosoever hateth his brother is a murderer: and ye know that no murderer hath eternal life abiding in him.16 Hereby perceive we the love of God, because he laid down his life for us: and we ought to lay down our lives for the brethren.17 But whoso hath this world’s good, and seeth his brother have need, and shutteth up his bowels of compassion from him, how dwelleth the love of God in him?18 My little children, let us not love in word, neither in tongue; but in deed and in truth.19 And hereby we know that we are of the truth, and shall assure our hearts before him.20 For if our heart condemn us, God is greater than our heart, and knoweth all things.21 Beloved, if our heart condemn us not, then have we confidence toward God.22 And whatsoever we ask, we receive of him, because we keep his commandments, and do those things that are pleasing in his sight.23 And this is his commandment, That we should believe on the name of his Son Jesus Christ, and love one another, as he gave us commandment.24 And he that keepeth his commandments dwelleth in him, and he in him. And hereby we know that he abideth in us, by the Spirit which he hath given us.

Another form of “know” is the Greek word eidó (Strong’s 1492) with the meaning to know or remember:

Matt. 25:10 And while they went to buy, the bridegroom came; and they that were ready went in with him to the marriage: and the door was shut.11 Afterward came also the other virgins, saying, Lord, Lord, open to us.12 But he answered and said, Verily I say unto you, I know you not.