You certainly can say she sings quite loudly and dances quite beautifully. Interesting, however, these would not suggest the same meaning: "sings quite loudly" would mean "moderately" loudly, it is useful to grade or show relative volumes. When people say "quite" in front of beautifully they are using it to intensify the beautifulness: "quite" is serving as "very" or "extremely".
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.