David was the son of Jesse of Bethlehem:- 1Sa 16:1 And the LORD said unto Samuel, How long wilt thou mourn for Saul, seeing I have rejected him from reigning over Israel? fill thine horn with oil, and go, I will send thee to Jesse the Bethlehemite: for I have provided me a king among his sons.
1Sa 17:12 Now David was the son of that Ephrathite of Bethlehemjudah, whose name was Jesse; and he had eight sons: and the man went among men for an old man in the days of Saul.
1Sa 17:58 And Saul said to him, Whose son art thou, thou young man? And David answered, I am the son of thy servant Jesse the Bethlehemite.
David came from Bethlehem:-
1Sa 17:15 But David went and returned from Saul to feed his father's sheep at Bethlehem.
1Sa 20:6 If thy father at all miss me, then say, David earnestly asked leave of me that he might run to Bethlehem his city: for there is a yearly sacrifice there for all the family.
Luk 2:4 And Joseph also went up from Galilee, out of the city of Nazareth, into Judaea, unto the city of David, which is called Bethlehem; (because he was of the house and lineage of David:)
Joh 7:42 Hath not the scripture said, That Christ cometh of the seed of David, and out of the town of Bethlehem, where David was?
The Bible does not say that David was born in Jerusalem, but since David's father came from Bethlehem, lived in Bethlehem, and that Bethlehem was David's city, it is safe to assume that David was born in Bethlehem and not in Jerusalem.