Luke does NOT name a Specific Governor of Syria.

 First in the actual Greek text of Luke 2:2 the word translated Cyrenius is the last word and Syria the second to last word.  And the word for "governor" is not a noun and should be more literally translated "governing".  In fact the most accurate rendering of the verse should be something like.

"This counting was first made during the governing of Surias Kureniou"

The last two words I chose not to transliterate and represent them as they are spelled in the Greek.

The only reason why Bible Skeptics insist this MUST be the AD 6 Census in-spite of all the ways it's nothing like that Census (Empire wide not local, and while Herod was still King or at least had been less then 2 years earlier) is the name of Kureniou.

In Josephus Quirinus is spelled Kurinios, which, like I would expect, uses more then one Iota.  Also there is no "e".  It is a much more plausible Greek rendering of Quirinus.

I'm not sure how early on this confusion started.  Maybe simply because Luke refereed to the AD 6 Census in Acts 5:37 people made the wrong assumption it must be the same Census.  Or maybe the translation of Luke into Latin played a key role in the confusion, when Translations of The Bible into modern languages finally began to happen after the reformation, they were greatly influenced by the Vulgate directly or indirectly, even the KJV.

But Tertullian in his against Heresies book IV chapter XIX simply states Saturninus was governor of Syria at the time without any acknowledgment that supposedly Luke identifies someone else as Syria's Governor.  That tells me that neither he or his readers had heard of the idea that Luke tells us who the governor was.  (Note, identifying Saturninus would fit it being the 8 BC Census).  And none of the 3 Lustrums of Augustus would have extended into the 19-21 AD Governorship of Gnaeus Sentius Saturninus, so the claim that Tertulian was referring to younger siblings of Jesus being born then doesn't work.  It was the earlier Saturninus who's administration of Syria coincided with a documented Roman Census.

To be exact, Tertullian said that Roman records proved the fact that censuses (he used the plural) were conducted in Judea when Saturninus was governor.  Also in his Apology to the Jews Tertullian clearly dates the Nativity to 3-2 BC saying it was 27 years from the deaths of Antony and Cleopatra.  But that is hard to reconcile with the Saturninus reference.

Since I'm contending that Kureniou doesn't mean Quirinus, what does it mean?

Below is how Cyrene and Cyrenian (of Cyrene) is rendered in various Greek NT verses.  Because these will be using 2 different Greek letters for o, lower case o is Omicron and capitalized O is Omega.

Matthew 27:32, Kurenaion
Mark 15:21, Kurenaion
Luke 23:26,  Kurenaion
Acts 2:10, Kurenen
Acts 6:9,  KurenaiOn
Acts 11:20, Kurenaioi
Acts 13:1, Kurenaios

It's rendered differently almost each time, in total 5 different ways, and Luke used all 5.  So that Kureniou is identical to none of them means little.  Interestingly the last one is almost identical to how the Strongs incorrectly claimed Cyrenius was rendered (Kurenios) with the only difference being the added Alpha.

The differences are all a matter of vowels and what the closing suffix should be.  All of them begin with Kuren just like Kureniou does.

Ending with iou is the same as how Luke renders Jesus of Nazareth in Luke 24:19 (Iesous tou Nazoraiou).

So perhaps Luke 2:2 wasn't referencing the Governor of a province at all but two provinces?

I could also note that when Luke identifies Pilate as Governor of Judea in Luke 3:1 he lists the name of the Governor before the name of the province.

Maybe the verse should just be translated as saying "when a Syrian was Governing Cyrene"?  Or perhaps that a Cyrenian was governing Syria.  Plausible translations are "during the Governing of Syria and Cyrene" or "during the Governing of Syria by Cyrene" or "during the Governing of Cyrene by Syria".  But I feel from everything I've observed above the best translation is "during the Governing of the Syrian of Cyrene".

An Atheist who is unlike me willing to consider the text hasn't been perfectly preserved should consider that a name is missing, that it's saying someone of Cyrene was Governing Syria.  Heck what Tertullian said you could use as evidence Saturninus was named in the texts he had.

The Roman Legion called the Legio III Cyrenacia was based for some reason in Bosra Syria;  Again I note the terminology of Luke properly translated is not necessarily identifying a person as Governor at all.

The last known exploit of this Legion before the time frame of The Nativity (from the timeline of the Legion Wikipedia has anyway), was being involved in a conflict between Rome and Nubia in Egypt in 23 BC.  The next time they show up is AD 7-11 when the Nikopolis fortress is established.

I'm thinking it's possible this Legion carried out the Census in Judea.

Given how often modern translations claiming to be directly translating the Greek are still influenced by the Latin translation, I was prepared to consider the Latin Vulgate perhaps the origin of this mistake.. But to my surprise the Latin doesn't mention Quirinius here, Jerome or whoever actually wrote the Vulgate did not recognize this as a Greek transliteration of a Latin name, it spells the name Cyrino.

haec descriptio prima facta est praeside Syriae Cyrino

According to Google Translate, everything preceding the two names at the end is, "This was the first President of the".  That is distinct from how the Vulgate does make references to Cyrene, but the main distinction there is using an E where that spelling has an I, something that isn't a difference in the Greek.  So the Vulgate translation is mistaken, but I find it fascinating that the educated Latin speakers who made it didn't see it as clearly a form of a specific Latin name.  This may possibly be similar to the version of the verse Tertullian would have read.