"to vibrate, tremble," early 15c., probably a frequentative of cwavien "to tremble, shake" (early 13c.), which probably is related to Low German quabbeln "tremble," and possibly of imitative origin. Meaning "sing in trills or quavers" first recorded 1530s. Related: Quavered; quavering.
1560s, in music, "eighth note," from quaver (v.). Meaning "a tremble in the voice" is from 1748.