So what is the problem?
Luke says this at the opening of Chapter 2:
In those days Caesar Augustus issued a decree that a census should be taken of the entire Roman world. (This was the first census that took place while Quirinius was governor of Syria.)
There is nothing symbolic here, this is just Luke providing us historical information to help frame what is to come in the account of Jesus (and possibly increase the historical soundness of that account too). So what does Luke say about King Herod? A heck of a lot less than Matthew. King Herod gets a single mention in Chapter 1:5.
(1:5) Herod + "After this" (v24, presumably not too long after?) + five months seclusion for Elizabeth + 3 months of Mary staying with Elizabeth. Then, after John's birth and naming, v80 states:
And the child grew and became strong in spirit; and he lived in the wilderness until he appeared publicly to Israel.
Chapter 2 begins as already mentioned above (In those days...). Since 1:80 seems to take us right up to John's public ministry, it seems safer to consider the "In those days" to go back, although how far back? The issue we have is Herod the Great's reign ended a decade earlier than Quirinius (full name is Publius Sulpicius Quirinius), who was actually taking over from Herod Archelaus, Herod the Great's (kind of) successor.
Apparently, most scholars have chosen to go with Luke getting his dates mixed up, making a small mistake here. Looking just at Luke, I wondered why this was the only option. Luke is writing I guess sixty years later and consulting various sources is doing the best he can to situate the significant chronology of events. Why could it not be that "In those days" referred to a decade or so after John's birth, during his childhood, John being an older cousin by ten years would work fine, right? That would also allow Herod (the Great) to be firmly out of the picture (i.e. "dead", Matthew 2:19).
Unfortunately for this little idea of mine, that does not go down too well for scholars trying to reconcile the Luke account with the Matthew account, which has Herod everywhere, right up to Jesus being at least one (assuming the 2-year old genocide ruling included a safety margin for big babies), probably older (plus the time waiting in Egypt for Herod to die). And according to Luke, Quirinius was already appointed governor of Syria by Augustus. That is the time-marker for the census preceding Jesus' birth.
So Matthew combined with Luke gives Herod as local ruler and Quirinius as governor of Syria as contemporaries.
So, if apologetics is your forté, I would be very interested to know what a good response is to this apparent discrepancy. Thanks!
UPDATE: Some technical responses can be found here, although they are more focussed on Luke than Matthew-Luke consistency.