If the Moon has a magnetic field and can possibly receive solar wind, then does it have an aurora? If Earth has an aurora, and Saturn has an aurora, then could the Moon possibly have an aurora?
Technically, yes, the moon does have an atmosphere that gets ionized by the sun. NASA: Is There an Atmosphere on the Moon?
our moon does indeed have an atmosphere consisting of some unusual gases, including sodium and potassium, which are not found in the atmospheres of Earth, Mars or Venus. It's an infinitesimal amount of air when compared to Earth's atmosphere. At sea level on Earth, we breathe in an atmosphere where each cubic centimeter contains 10,000,000,000,000,000,000 molecules; by comparison the lunar atmosphere has less than 1,000,000 molecules in the same volume.
Pretty thin soup. Sodium and potassium get kicked up by photons hitting the surface. There's also argon, helium and other minor components. As shown in this figure from the first link: The sodium in that atmosphere does glow; (Rayleigh unit definition). At least some of the glow is due to full fledged ions rather than charge-neutral fluorescence:
It is confirmed that the ions include oxygen, sodium and potassium.
So, whether by ionic or fluorescent emission, the moon's atmosphere does glow. Unless one wants to argue about the glow not being restricted to the poles by a magnetic field, or airglow, that glow counts as an aurora.
Auroras require an atmosphere, which the moon does not have.
Auroras occur as solar wind and other particles that interact or are associated with Earth's magnetic field are funneled by the magnetic field and accelerated until they collide with atoms/molecules in Earth's atmosphere. These collisions cause the atoms/molecules to ionize or excite the the subsequent recombination/de-excitation of the electrons is what causes the light seen in the auroras.
Without at atmosphere, though, there are no particles to ionize and excite to create an aurora.