How many moons are there on Mercury?

None.

As described in “The Data Book of Astronomy”, by Patrick Moore, there aren’t any moons on Mercury. In 1974, prior to the Marine 10’s Mercury flyby, it was thought that Mercury had a satellite, due to extreme ultra-violet radiation coming from the planet’s surroundings. Eventually, it was discovered that the supposed moon was in fact a simple star: 31 Crateris.