Several verses come to mind with this question:
Rom 1:19 Because that which may be known of God is manifest in them: for God hath shewed it unto them.
Job 11:7 Canst thou by searching find out God?
Utilizing a musical metaphor, there seems to be, according to the Romans 1 verse a resonant "God Chord" within man that if generally revealed truth be followed without corrupting, it would lead one to at least discover the "eternal power and Godhead" of God.(vs 20). And of course when the truth of God’s general revelation is encountered that resonant chord within begins to vibrate to the tune of "Holy, Holy, Holy" and unconverted man then begins to run as fast and as far as he can from this truth. So it seems to me that to try to "prove" that the God of the Bible exists is a futile effort from the outset. The answer to the Job verse is, of course...No! God must reveal Himself to be the God of the Bible. And....."Amazing Grace"....He does just that!
William Dembski writes in the introduction of his book MERE CREATION: "To say intelligent causes are empirically detectable is to say there exists well-defined methods that, on the basis of observational features of the world, are capable of reliably distinguishing intelligent causes from undirected natural causes." He goes on to say..."Intelligent design detects intelligence without speculating about the nature of the intelligence"
Man even runs away from the truth of "intelligent design" which doesn’t even attempt to identify the designer. Rom 1:20 "For the invisible things of Him from the creation of the world are clearly seen...." Man wants no part of, even, an unidentified "intelligent designer" even if empirically detectable.
But to the one for whom you would desire to prove that the "God of the Bible exists.." you pray for that one, you preach the Word and be prepared in season and out.. that: "...the God of our Lord Jesus Christ, the Father of glory, may give unto (that one) the spirit of wisdom and revelation in the knowledge of Him:.." Eph 1:17 and then you can say to that one with Paul, "And you hath he quickened who was dead in trespasses and sins:.." Eph 2:1 It is truly by GRACE that we are saved!
Then...he needs no proof...Christ says to him " I was made manifest to them that asked not after me". Rom 10:20b I KNOW whom I have believed and am persuaded that he is able...." II Tim 1:12
This question could be the basis of a good sermon, by a good preacher, or, at least, a good song:
I sought the Lord, and afterward I knew
he moved my soul to seek him, seeking me;
it was not I that found, O Savior true;
no, I was found of thee.