A person who is cut to the quick is one whose feelings have been deeply hurt. The expression provides an illustration of the original meaning of quick, which comes from the Old English adjective cwic, meaning “alive”.
A cut which penetrates the quick is one which goes through the skin and enters the painfully sensitive living flesh, as found beneath the nails.
Source: The Penguin Wordmaster Dictionary, Martin Manser and Nigel Turton, eds., 1987.
A better illustration might be, cutting through a callous to reach the living, sensitive skin beneath. (But does a callous consist of dead skin? I dunno.)
Another example of quick meaning “living” immediately comes to mind: quickening, the first time a mother-to-be feels her baby move inside her. The quickening indicates that the baby is alive — i.e., cwic.