It thunders at night, there are stars and moons in the sky, and it doesn't rain. Is this a precursor to an earthquake?

No, this is because the place where it rains is not in your place, but the light of lightning is very strong and can cover many places. Thunder is so loud that it can be heard far away, and your place is sunny, so you can see both lightning and thunder, as well as the beams of stars and the moon. This is a normal natural phenomenon.