Quavery describes a voice that shakes as you speak. People sound quavery when they are nervous, upset, close to tears, or sometimes when they are old. You say “a quavery voice” or “she sounded quavery.” It comes from the verb quaver: if your voice quavers, it shakes; quavery is the adjective for that kind of voice. It is a little literary — in everyday talk many people just say “shaky” or “his voice was shaking.” A story might choose quavery. The sense here is the shaking voice, not a music note.