I was always under the impression that the answer was no, but Rufus Fears refers to it as a normal part of becoming a province, and I found multiple Catholic references to a historic census.
Yes, there's corroboration, but not a close match for what Luke describes. And not only does the external evidence contradict the timeline based on the census: Luke contradicts it himself at a different point.
###Luke's census
Luke's version of the census has three key features:
There are census that could arguably be a loose match for point 1, but they have nothing to do with Judaea; there's corroboration for a census fitting point 2, but it's too late to have anything to do with Jesus' birth; and point 3 is intrinsically nonsense.
On point 1, Augustus' Res gestae refers to three censuses of all Roman citizens, taking place in 28 BCE, 8 BCE, and 14 CE. In that period 'citizens' meant mostly Italians, plus citizens of Roman colonies established outside Italy, plus people who had citizenship because their parents were citizens. That is, these censuses didn't include the provinces in general. They didn't include newly incorporated regions that didn't enjoy citizenship. And the ones in 28 BCE and 8 BCE definitely didn't include Judaea, because Judaea wasn't even part of a Roman province until 6 CE.
On point 2, we have two external sources corroborating a census of Judaea when it became part of the province of Syria in 6 CE. They are Josephus, Jewish antiquities 18.1; and an inscription from Beirut, CIL (Corpus inscriptionum latinarum) iii.6687, at line 9. Josephus gives a clear account of how the ethnarch Herod Archelaus was expelled by the Romans and Judaea was placed under direct Roman rule, and records that as a result the governor, Quirinius, came to Judaea to assay the new territory for taxation and to dispose of Archelaus' assets.
Point 3 is simply nonsensical because 'each person's city' isn't a coherent idea. Luke 3 has forty generations separating Joseph from king David: is Joseph supposed to visit the hometown of every ancestor in that lineage?
###The reason for the 'everyone went to their own city' weirdness
It's pretty clear that the silliness in point 3 was motivated by a dissonance between Jesus' actual origins, and a theological expectation about where the Messiah was supposed to come from.
All four canonical gospels are clear that Jesus came from Galilee, not Judaea. (Only Matthew makes any suggestion that he had origins anywhere else, and his account of the family's movements doesn't hold water any more than Luke's.) But it seems that in some quarters, at least, there was a messianic expectation (based on the Hebrew prophetic book of Micah, 5.2, which Matthew quotes) that the Messiah was supposed to be born in Bethlehem.
John 7.41-42 reports a dispute prompted by this exact dissonance.
In light of that, and in light of Matthew's and Luke's contradictory accounts of the family's movements, it's pretty clear that the stories in Matthew and Luke were each concocted to explain away the dissonance in a different way. So Matthew has Jesus' family living in Bethlehem, then later moving to Galilee; and he explains the move as being because Jesus was pursued by Herod Archelaus (r. ca. 4 BCE to 6 CE). And Luke has Jesus' family moving from Galilee to Bethlehem and then back again; and he explains their movements as being because of Quirinius' census (in 6 or 7 CE).
###The problem with Luke's internal chronology
Finally, Luke 3.1 sets Jesus' baptism in the 15th year of Tiberius' reign. This was a very conventional way of reporting dates in the eastern empire. In 3.23 he tells us Jesus was 30 at the time. This would put the baptism in 28-29 CE, and Jesus' birth in 3-2 BCE -- at least seven years earlier than in Luke 2.
It appears the author of Luke didn't have a firm grasp of the timeline. Luke 1.5 refers to the ruler of Judaea before Quirinius as 'king Herod'. But king Herod the Great died in 4 BCE or shortly after; his successor was Herod Archelaus, an ethnarch (not king). It looks as if the author of Luke was (a) aware that Jesus was born near the end of the reign of Herod the Great, but (b) unaware of the distinction between Herod the Great and Herod Archelaus. I conjecture that this error may have motivated his linking Quirinius' census to Jesus' birth, even though the events were separated by several years in reality.
Given the touchy nature of the subject I need to add that all of this is the mainstream view among biblical scholars, and entirely uncontroversial (except perhaps my conjecture in the last paragraph: the scholarship may have precedents for that too, but I don't know them). The most detailed work on the subject is Raymond Brown's classic study The birth of the Messiah (1977, 2nd edition 1993), which makes most of the points I've made here.
And a final note on the various Herods that appear in the relevant sources: there are three to be aware of --