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.
I would also point out that the census being mentioned by Luke is just one bit of evidence that Luke used Josephus as a historical reference. The fact Josephus has John the Baptist being executed in 36 c.e. and Jesus starting his ministry that same year is another clue, since at that point, Jesus would have been 30, as described by the gospels.
I have seen this line of reasoning.. about 'before'. However, there are a number of problems with it. The first one, there is no historical record of any census of Judah happening by the Roman empire before 6 c.e.
Before 6 c.e. , Judah was a country that paid tribute to Rome, but was not taxed directly. The census was for tax purposes, and Rome did not directly tax Judah until 6 c.e.
That is trying to distort the words written down to a large degree, and then ignore the history, laws and structure of Rome at that time to try to make an excuse.