Do the gospel writers disagree on who was ruling when Jesus was born?

Only Matthew and Luke mentions the birth. Matthew says that Herod was king of Judea, and Luke provides the detail that Caesar Augustus was emperor, and Quirinius was governor of the province of Syria. None of these details contradict one another.

Matthew 2:1 Now after Jesus was born in Bethlehem of Judea in the days of Herod the king, behold, wise men from the east came to Jerusalem

Luke 2:1-2 In those days a decree went out from Caesar Augustus that all the world should be registered. 2 This was the first registration when Quirinius was governor of Syria.

Note that is says “when Quirinius was governor of Syria”, not when he was governor of Judea. This does not refer to the 6 AD census by Quirinius. That 6 AD census is however referred to in Acts 5:37.