Statistically there's no increase in quakes around full or new moons:
http://seismo.berkeley.edu/seismo/faq/planets.html
There are some slight statistical increases with certain kinds of quakes relating to DAILY tides, but not to moon phases.
However there is an OVERWHELMING desire of people to believe such a moon phase relationship exists, for reasons that are somewhat obscure to me.
A major driver of the myth is noted seismological fraud/liar Jim Berkland, a regular guest on Coast-to-Coast, though.
PMS is easier to comprehend if its not random.
For some people it may be a desire, but it also seems logical to people who have a bit of scientific knowledge.
But the problem, of course, is they forget that the moon is not alone. The sun is there, too, so we need to look at the combined effect, which is more closely related to the tides. This slight correlation to some aftershocks is mentioned in the Berkeley link.
But the water tides lag behind a bit due to several factors. I am not sure if the 'rock tides' lag in a similar amount. I doubt it. If not, it would be interesting to look at that for correlation if it hasn't been done already.