First, Luke 2:1 says KyrhnioV (pron. Kyrenios), which means Quirenius in Latin. No confusion there.
Second, the NT tells us that Jesus was born during the life of Herod the Great. This Herod died in 4 BC.
Third, another Herod (Arhelaus), succeeded Herod the Great (Archelaus' father) and ruled as an etnarch of Samaria, Judea and Edom between 4 BC and 6 AD, when he was exiled to Gaul and Judea was turned into a Roman province, and Quinirius was tasked with compiling a census of the region.
Luke clearly states that it was under Quinirius (not Quintilius!) "everyone was traveling" (v. 2:3) in order to register for the census ordered by Quintilius. It then says (verse 2:4) literally "Now went up also Joseph...and Mary, the one having been engaged to him, was pregnant [εγκυω]," in order to register.
There can be no doubt when this happened. There was no census in that aprt of the region untl 6 AD.
Yet Matthew 2:13-23 speaks of Jesus being alive when Herod (the Great) was still living (before 4 BC). The problem is that Luke contradicts himself (1:5) and is agreeing with Matthew!
Fourth, according to Matthew, Herod ordered the slaughter of the Innocents. Tere is no record of any kind that this happened, not even by Jospehus who wa sin a habit of describing thing about Herod in great detail.
Under either of these scenarios, SOMEONE served twice, and under either of these scenarios, Quirinius could EASILY have been responsible for the census There would have been no need for census until Judea was made a Roman province, and this happened in 6 AD. Matthew's accounts says Jospeh and Mary and her Son went to Egypt but were told to go to Nazareth after Herod died.
Accoridng to Luke (2:1), however, they went to Nazareth to be registered because everyone went to their respective cities. No angel calling them out of Egypt!
Accrding to Luke 1:5 a completely different sotry develops. We have a clear inter-author and inter-bible contradiction or, to use a PC term, disagreement.
If you had read the studies with a fair heart, it will become abvious that Quintilius ruled in various capacities over that region for about 20 years prior to him being formally assigned as potentate. In many of those capacities he was a coruler without much impediment. I see no significant compromise.