I've never heard anyone classify Jesus' birth as either Old Testament (OT) or New Testament (NT). If anything, he was born in the Intertestamental Period, which basically is just what it sounds like--the time between the two testaments.
Because the terms "Old Testament" and "New Testament" refer to a set of Sacred books, rather than a distinct time period, it may not make a lot of sense to talk about Jesus being born in either the OT or the NT. The account of Jesus' birth in the Gospels of Matthew and Luke are, of course, in the New Testament. But those accounts were not written during Jesus' lifetime and probably not in the lifetime of many who were eyewitnesses to Jesus' earthly life and ministry.
If we refer to the OT Period and the NT Period, however, scholars think that Jesus was born sometime in the last decade before the "Common Era" (Anno Domini, Year of our Lord). He is said, in the gospels, to have been born during the reign of Herod the Great, and that particular Herod seems to have died in 4 BC.
Christianity and the New Testament was formed during the 80 years or so after Jesus' death and resurrection. The Acts of the Apostles makes it clear that Jesus' first followers thought of themselves as Jewish. And throughout Paul's lifetime (he died in the mid-sixties) it was a controversial question whether Gentile converts would consider themselves Jewish or not.
Both Christians and Jews today think of themselves as heirs to the People of Israel described in the Christian OT and the Hebrew Bible (much of which overlaps). It is the hope of many people of good will that our common heritage might become a source of greater dialogue and increased understanding between the faithful of these two great religions.
And while we don't really know the precise date of Jesus' birth, we celebrate 2000 years of the Incarnation of God-with-us, Emmanuel, every December 25. Merry Christmas my friends, and may God's holiness fill your house whatever holy days you celebrate at this time of year.
