. . . between the birth of Christ in Luke and in Matthew.
According to Luke:
Luke 2 say that Joseph and Mary are living in Nazareth, but because of a census when Cyrenius (AKA Quirinius to give him his Roman name) was governor of Syria they had to travel all the way from Nazareth to Bethlehem.
There is no room in the inn, so they stop in a stable and Jesus is born and laid in a manger.
They then travel on to Jerusalem, then back home to Nazareth where Jesus grows up.
According to Matthew:
Matthew 2 says that Jesus is born in Bethlehem, apparently in the family home, during the reign of Herod the Great.
The three Magi come to their house and visit, then depart. Joseph is warned by an angel that he must flee.
So Joseph, Mary and Jesus flee to Egypt where they live for some time until Herod the Great dies and they consider it safe to return.
So they are heading home but discover that Herod Archelaus rules the area including Bethlehem, so they head north to Galilee and settle in Nazareth. This is further evidence that they had lived in Bethlehem, not Nazareth.
There are several problems with this.
Date of the birth - Quirinius/Cyrenius did not become governor of Syria (and Judea) until after Herod Archelaus was removed from power. One of these accounts has to be wrong.
Census - people did not travel huge distances for a census. Why does Luke say that they did?
Luke says that Joseph and Mary lived in Nazareth and returned via Jerusalem to Nazareth. Matthew says they lived in Bethlehem fled to Egypt and then moved to Nazareth. Which is right and which is wrong?
If they were afraid of Herod the Greats son, Herod Archelaus such that they were afraid to return to Bethlehem, why were they not afraid of Herod the Great's son Herod Antipas? Why not go to Syria, or the Decopolis cities where they would not have any of Herod the Great's sons over them?
If one of these accounts is seriously wrong, then what does that say about the rest of that particular gospel?