Basic Info
The Quantum Moon, referenced by the aliens as the "Wandering Moon" travels from planet to planet when not being observed. It exists in four different locations, with a fifth being alluded to by numerous inscriptions. This fifth position only exists within the planetary system for a short period of time at the very start of a session, making it very difficult to trigger. Its path in this position does not appear to be in relation to any other body, and quickly takes it out into deep space until the sun can no longer be seen from it.
Landing and Surface
The best way to land is to use the probe to take a photo (this is explained here in more detail), in order to lock the moon in place. Then land as normal. The surface is shrouded in a thick layer of fog, and is barren except for a sign post and a hut. Both of these landmarks move when you look away from them.
The Physics
Now for the complicated bit. The moon exists in a quantum superposition, where it exists only as a probability across every location. When you look at it, the probability disappears and the moon is in only one place. Looking away from it puts it back in the superposition, and hence is why it moves. Thus the moon can be represented as a wavefunction. Taking a photo of it with the probe forces it into one position, locking it in place even when you yourself are not directly looking at it. This is basically Schrödinger's moon.
Trivia
The moon may be where the phase shifting sculpture came from. The sign post is the same one as is in the village on Timber Hearth.