Yes, clothes moths will happily feed on feathers, as I saw one time when case-making clothes moths were destroying the feathers on a box full of Kachina Dolls from the Southwest. This is simply the role of these decomposers in Nature. Likewise, the hair of stuffed mammals also would be susceptible to etierh of these groups of insects. 

What the museums do is to remove the specimens on display periodically and treat them in some manner that will kill any insects or their eggs that may be on the mounts. This may be heat treating, which could be done quickly, freezing, which would take much longer, or the use of a sealed chamber and a fumigant, and the two easiest fumigants today would be either PDB (paradichlorobenzene as in moth crystals) or vapona. The vapona is now the Nuvan Prostrips, which are directly labeled for "museum collections", which I believe these stuffed mounts would fall into. At this time we are, frankly, blessed with the ability to use the Nuvan strips in our industry. We know the benefit they provide in bed bug control, but if the quail or elk can somehow be placed within a sealed surrounding of thick plastic sheeting the Nuvan strips can be placed inside for a couple of weeks to ensure the kill of any moths, their larvae or pupae, or their eggs. 

Vapona strips have long been used in insect collections without causing any change in the appearance of the insects, including fragile butterflies. It should have no effect on the feathers or fur of stuffed animals, and it also should leave no odor behind, which PDB may tend to do. Having worked in insect museums for decades I kind of like the slight odor of PDB - it makes me feel at home - but some people may not be as tolerant of it as I am. If this customer does opt for moth crystals have them use PDB rather than naphthalene, as naphthalene is not as effective at actually killing insects or their eggs.