Yes, of course. The hair on your face is hair just like on any other part of your body - including your head. It can be colored by the same products you use to change your haircolor. There are some differences, obviously. The one most readily apparent to everyone is texture.
 
Facial hair tends to be much coarser than the hair on the head, and as a result can be more resistant to haircolor. The facial hair can also be less dense than the hair of the scalp and color application must therefore be done much more carefully to minimize the staining of the skin underneath.

 ~ A