Nope. There have been more than a hundred studies on the alleged connection between moon phases and behavior, but none have found a statistically significant correlation. Anecdotal evidence from police and emergency room workers that behavior changes is simply confirmation bias.
If it were really true that lunar gravity caused strange behavior, we'd expect to see the biggest spike in crazy behavior at mid-day during the new moon, when the tidal forces from the Sun and Moon are highest. We don't.