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Visual Doppler? No, that's just not going to happen.
First off, as Fossil points out, the moon's velocity through space is entirely inadequate to produce a gross color shift. A sensitive spectrometer might see a very, very tiny wavelength/color shift, but to go from eg. blue to green would require MUCH more speed, on the order of 20% the speed of light. (See http://www.asterism.org/tutorials/tut29-1.htm )
Secondly, orbital velocity will not produce any significant spectral shift. Radial velocity (change in distance over time) is what you want. The moon's distance does vary during the course of it's orbit (by +/- 12,000 miles), but this occurs over many days, resulting in a rather low radial velocity that peaks somewhere in the low 1xx miles per hour range. The redshift this would produce would be completely, utterly, absolutely undetectable by the human eye.
One more point: Since the moon it gravitationally locked, with one side more or less facing the Earth at all times, you're not going to get much rotational velocity either.
Also consider that ALL the wavelengths emitted by the object are shifted. Depending on the direction of travel, ultraviolet or infrared will be shifted into the visual spectrum. If the object is not strongly colored in the first place (the moon is a pretty dismal dark grey with few emission peaks), there may be little apparent change.