I found a Dan Wallace article that talks about this, and he exposes some explanations as being failures. And this is how he concludes the article...
"In conclusion, facile solutions do not come naturally to Luke 2:2. This does not, of course, mean that Luke erred. In agreement with Schürmann, Marshall “warns against too easy acceptance of the conclusion that Luke has gone astray here; only the discovery of new historical evidence can lead to a solution of the problem.”
I'm wondering however, is this even possible? Let's try to theorize a future discovery that would vindicate Luke! You would need 2 right? First proof that Quirinius was Governor in 4 BC, and then proof that there was a 4 BC census in Judea! Am I wrong, is it possible to theorize just 1 future discovery to vindicate this 'Error?'
No, a single discovery is quite possible. It would just have to be very specific.
What you should remember firstly, is that in the close of Herod the Great's reign, he was out of favour with Augustus.
Herod's reign ended chaotically. Matthias and Judas, two rabbis in the temple, incited an uprising to remove the Golden Eagle he erected over the Temple. He executed his two sons Aristobolus and Antipater, who he felt were plotting against him - this led to Augustus' snide remark that it is better to be Herod's pig than his son.
Herod had also shown an independant streak in making war on the Nabataeans and trying to control the incense trade, while at the same time building a number of fortified positions like Masada and Herodias, that Rome did not particularly like in her vassals. This was afterall a strategic frontier defending Egypt and Syria's flank from Parthia.
Augustus tended to annex troublesome vassals and to organise their realms into provinces, as he would later do after Herod's son Archelaus' mismanagement.
It is clear that in the closing years of Herod's reign, his independance was curtailed a bit by the Romans, with annexation likely a looming threat - which is why his sons were so unsure of Augustus accepting the provisions of Herod's will.
Secondly, remember that we are dating Quirinius' census by Josephus. Now Josephus is a known corrupted source - for instance the clear fact that the Testamonium Flavium had been under the pen of a redactor, as the extent version is different then early Christian references to it; and the addition of extra material, like Paulina's scandal in the middle of an account of Pilate's career.
Josephus is mostly taken as accurate, but it is not hundred percent.
So, if we keep these two points in mind, a single discovery could account for the two gospels' differences.
If we discover an account or decree of Augustus, that appointed Quirinius to oversee a provisional or preliminary investigation of the annexation of Herod's realm, then we are golden.
For a census done at this time would be the first Indiction of Judaea, with the second Indiction being the 6 AD census - with Josephus or a later redactor confusing the two, as Judaea wasn't annexed at that time afterall. It would also explain why Galilee would fall under the census; which doesn't fit the 6 AD one, where it was outside the Roman province.
For Quirinius to do so, he would likely be appointed the Syrian legate, thus supporting his 'governorship', and the Syrian governor at that time is unknown (although likely Calpurnius Piso, and Quirinius being legate of Syria and Galatia that this would entail is unlikely - Augustus did not usually allow such concentrated power). As it would be a short term appointment, that came to nothing in the end, it may be overlooked by Tacitus and others on account thereof.
Alternately, an early Josephus that lacked the passage on Quirinius' annexation, or one that placed the passage earlier in the narrative, would also allow a lot of leeway to reconciliation of the Gospels. It would not confirm them though. This is again possible, as Josephus is known to be a little corrupted (although this would itself be a significant discovery in that Antiquities would thus be even more corrupted then previously thought).
A discovery of an early Matthew lacking the birth narrative, similar therefore to Mark or John, would also support Luke being in the right - but this would not help supporters of Biblical accuracy much.
More complex arguments based on multiple new discoveries, text or inscriptions, is far more likely therefore, if we are ever to reconcile the two Gospels' accounts.