The planet Mercury does not have a moon. At one time, astronomers from NASA thought the planet Mercury might have a moon, but research proved that theory to be incorrect.
When astronomers thought Mercury had a moon in the mid-1970s, it was because of data about an unusual amount of radiation that had been received from the Mariner 10 mission. This radiation turned out to be created by a star, not a moon.
One theory is that Mercury might have, at one time, had a moon that broke apart, leaving debris to crash to the planet. This theory could be reasonably supported by the large amount of impact craters covering the surface of the planet.