The apparent colour of a body in space, such as a moon, depends upon a number of factors:
The spectrum of light emitted by the light source (the system's star or stars).
The light absorption and reflectance properties of the body's surface material.
The light absorption properties of the atmosphere (if any) of the body upon which an observer is located.
The spectral sensitivity of the optical apparatus observing the body.
Assuming that we want to keep 1, 3 and 4 constant, that means changing 2: the light absorption and reflectance properties of the body's surface.
With respect to Earth's moon, the lunar regolith is grey, absorbing some incident light and reflecting some, at a level effectively constant across the visible light spectrum. Mars' surface is covered in iron-rich substances that reflect more red light than other parts of the visible spectrum, hence its generally red appearance.
By surfacing a moon with a material such as olivine, it would be possible to give the body a green appearance.
There is an additional way that the apparent colour of a body could be changed. This involves red/blue shift, but an orbiting body would not have any significant doppler shift, that would likely be reserved for rogue objects approaching or retreating from the observer.