There have recently been two large earthquakes in California and Iran. They occurred at a time when Earth was close to perihelion. It was also close to a New Moon. Could there be a connection? Is there any correlation between earthquakes and astronomical movements?

Yes there is. Just as the sun and moon pull the oceans toward them creating a bulge of water one place and lower in the other, so their gravitation disturbs rock at a fault. So if a mass of land is under stress, and wanting to slip, it is more likely to do so when the sun or moon is pulling in the same direction. I am not sure the phase of the moon has much to do with it, just the position in relation to the strained rock.

Thanks for your reply, Labman.
The phase of the moon might be relevant because at New Moon the lunar gravitational pull is exerted in the same direction as the Sun's and so the effect would be greater than at a gibbous phase. However, I was wondering whether, statistically, earthquakes, along their fault lines, are essentially random and unpredictable events or whether there has been shown to be a triggering effect by tidal forces.