There are a number of mistakes in your recitation.
First, there is no evidence that Quintarsis was a governor of Syria more than once. The entire argument is a 'well maybe' , based on an inscription that had the person it was referring to totally erased by time.
The next big problem with the entire story of Luke is that Galilee was not part of the Kingdom of Judah, but was being ruled by Antipas. It was not subject to taxes at the time. Judah was, because it's ruler was just replaced, and put under direct roman control.
So, Luke's entire story , with Joesph and Mary coming from Nazareth, and Mary being that close to labor is unlikely to the extreme.
It is, however, some bit of evidence that the writer of the Gospel of Luke/Acts did use Joesphus as a historical source.