It should be, 'He enjoys nothing but listening to music'. I disagree with Jessica on this being an odd way to phrase. It sounds very natural to me.

Thanks, Sylvia Ujwary and Jessica.

What Jessica said was correct. Another option: He doesn't enjoy anything except listening to music.

"He enjoys nothing but listening to music" Would be technically correct, but it's an odd way to phrase what you're trying to say. I provided a few more options below that sound more natural. "He only enjoys listening to music." "The only thing he enjoys is listening to music."