We CAN determine the year of his birth to a narrow range, as I laid out in another question. Regarding the birth itself, the Gospel of Luke places it during the census under Quirinius, but the only such time known for sure is 6 A. The Gospel of Matthew places it during the reign of Herod the Great, who died in 4 B. However, it's worth noting that the Gospel of Matthew records two occurrences around Jesus' birth. Here's what I wrote: The year of Jesus' birth is uncertain, but we can narrow it down to a likely range based on the information given in the Gospels. The Gospel of Luke states that Jesus was about 30 years old when he started his ministry. The first is the star of Bethlehem noticed by some astrologers (Magi). The biggest clue, by far, appears to be the death of King Herod the Great. Chadwick wrote that the historical record places Herod's death at the end of March or beginning of April in 4 B. This date is confirmed by both the mention of a lunar eclipse before Herod's death and the date his son was deposed by Caesar Augustus. Luke placed this event in the sixth month — which at the time was from mid-to-late February to mid-to-late March. Both those fixed events fit together to confirm Herod's death.
We saw in the previous post that Josephus said Herod died between a lunar eclipse and Passover.
He tells us that when Jesus was born, Herod panicked and had all the baby boys in Bethlehem killed.