Short answer

No. Matthew and Luke tell two stories that cannot both be true. In Matthew, Jesus is born while Herod is king. The family flees to Egypt and stays there until Herod dies. On the way back they avoid Judea, because Herod's son Archelaus rules there, and settle in Nazareth. In Luke, the family already lives in Nazareth. They travel to Bethlehem for a Roman census taken when Quirinius governed Syria. About forty days after the birth, they go home to Nazareth. There is no flight to Egypt. The dates clash too. Josephus, a Jewish historian of the first century, says Quirinius took that census after Archelaus was removed. That was about ten years after Herod died. The Qur'an affirms that Jesus was born of a virgin. It is the Gospel writers' details that disagree.

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