I'm not the most professional on the topic but here's my two cents.
Velvet comes in many different forms, varying in thickness, stretch and appearance.
Some of them shine, some of them don't. Coats usually call for thicker, sturdier fabrics, but upholstery fabric is a bit much. Situationally it can be good but I dont personally think you need it for a coat. Cotton velvet seems to have the most mild shine imho., for a preview just google cotton velvet coat. (make sure you get one that doesnt stretch)
Just think, what do you look for in a coat; thickness, rigidness to hold its shape and able to be worn with comfort.
This doesn't mean you have to limit yourself to velvet, there are all kinds of fabrics that can suffice that criteria, just research some suiting fabrics. And also consider velvet is harder to maintain; stuff like it cannot be ironed, it has to be steamed and does call for different equipment to work with. (http://www.sewmamasew.com/2015/02/ty...w-with-velvet/
Whatever you choose to work with, good luck!