Where can you find the verse in the Bible that says God knows my name?

Answer # 1

The exact words "God knows my name" are not in the King James version.

Nor are the words "knows my name" or "knows your name".

Answer # 2

The following verse often speaks to readers in a very intimate and personal way:

But now thus sayeth the Lord that created thee, O Iacob, and he that formed thee, O Israel; feare not: for I have redeemed thee, I have called thee by thy name, thou art mine. (Isaiah 43:1, 1611 KJV)

The motif of God knowing us each by name is also echoed by Jesus when he uses 'gate' and 'good shepherd' as figures of speech referring to himself:

whoever enters through the gate is the shepherd of the sheep...the sheep hear his voice, as he calls his own sheep by name and leads them out...he walks ahead of them, and the sheep follow him, because they recognize his voice... I am the good shepherd, and I know mine and mine know me... (John 10 NAB)

The passage from Isaiah, I have called you by your name, you are mine, is also well-known in contemporary Catholic liturgies through the song You Are Mine, written by the songwriter David Haas who is prominent among Catholic musicians.

REFERENCES

The Bible Museum. The Bible: 1611 King James Version: 1st Edition, 1st Printing, (Goodyear, AZ: The Bible Museum, 2006).

Confraternity of Christian Doctrine. The New American Bible, (Iowa Falls: IA, World Bible Publishers, Inc. 1991).