Actually, yes, but indirectly
Well, the tidal forces produced by the Moon's gravity are not enough to distort the Earth's crust enough to produce any significant displacements of the crust, and produce earthquakes. The only significant effect would be producing tides on the ocean. Remember the crust is rigid, whereas water is not.
However, the Moon's gravity does warm up the Earth's core up to some degree, due to the continuous squeezing and stretching of the Earth (Most of the internal heat comes from radioactive decay and heat from the formation of the planet). It is this warmth, that drives tectonic plates, and thus ultimately, earthquakes.
EDIT: Crustal tides are indeed produced by the Moon's gravity. In fact, this effect causes New York to rise and sink by 14 inches in a day. But this effect is too smooth. You cannot perceive this effect with the naked eye. Crustal tides are not violent, rapid events. They occur gradually throughout the day. On the other hand, Earthquakes are produced by many reasons, most common being the sudden release of pressure and energy by tectonic plates, in just a fraction of a second. This sudden release of energy triggers vibrations in the plates, which we call Earthquakes.
Lunar gravity is simply not strong enough to trigger massive and rapid disturbances in the crust to create Earthquakes of sufficient magnitude. Even if it did happen, it would mostly be restricted to Magnitude 0-2 on the Modified Mercalli Scale, which can barely be noticeable to humans.
The only place where gravity/tidal forces can trigger powerful earthquakes would be a moon of Jupiter, Io. Io's interior, which is already under massive stresses from Jupiter's gravity, can find itself further distorted by combined gravitational forces from Ganymede, Callisto and Europa, which can trigger massive earthquakes (Ioquakes?) and of course, volcanoes.