Where is a Hollow Object's Center of Gravity?

If by "a hollow object" you mean a spherical shell -- a theoretically perfect sphere Moon, with a perfect sphere of material removed from the center: The center of mass, and therefore the center of gravity remains at the geometric center. Ref Newton's Law; Bodies with spatial extent.

The body of the moon is plastic; Meaning the gravitational forces are strong enough to reshape the material... it basically flows into a sphere. So if you magically removed a small amount of the center, it would collapse into a smaller sphere. If you removed most of the center, then the "thin" shell may or may not be strong enough to support it's shape.

Inside a hollow spherical shell, there is no net gravitational force. So you would be in a zero-G environment everywhere within the shell. From outside the shell, the gravity is the same as if all the remaining mass was located at a point at the geometric center.