I had to look this one up!
Alive" is the original meaning of quick. The quick and the dead did not refer to gunslingers in the Old American West, but instead refers to "the living and the dead" as in the Bible, Acts 10:42. The current meaning of quick, "rapid", did not emerge until the 13th century.
It's archaic, but "alive" was the original meaning of "quick".
Now, here comes the interesting part. The flesh under the nail is called quick because the nail, the hard part, is dead, but the flesh is "alive," thus the origin of "quick" as the flesh under the nail:
So in modern English we would say ´She bit her nails to the flesh´ or ´bit until the bone´.