No. Mercury has no moons, and it has no rings either. NASA notes that Mercury is the smallest planet in the solar system and the closest to the Sun, only slightly larger than Earth’s Moon, and it circles the Sun every 88 Earth days.
Mercury and Venus are the only planets in our solar system without a natural satellite. Earth has one moon, Mars has two, and each of the four giant planets has more than a dozen.
Source: NASA Science: Mercury Facts
Image: Mercury in true color, NASA/Johns Hopkins University Applied Physics Laboratory/Arizona State University/Carnegie Institution of Washington, public domain, via Wikimedia Commons.