This is about a supposed historical inaccuracy in Luke. There are various solutions to this supposed problem but we’ll cite here the conclusion of a paper in 1984.
V. CONCLUSION Many censuses were taken in the Roman empire during the time of Augustus, and there is no reason why Herod might not have been asked to take one, especially in light of conditions near the end of his life. Since censuses were carried out locally, local customs were regarded and Palestine was a delicate area. Quirinius may or may not have been governor of Syria at the birth of Christ in 5 B.C., but this is irrelevant since Luke 2:2 states that the census during which Jesus was born was the first one, before the more well-known one taken by Quirinius in A.D. 6-7. This first one was “in the days of Herod the king.”
This is from Liberty University SOR The Census and Quirinius: Luke 2:2 by Wayne Brindle 1984