He is apparently a criminal, with some history with Rooster.
In True Grit Quincy and Moon are staying at a dugout and getting ready to give Ned Pepper,Mexican Bob, and Tom Chaney horses so they can be used in a robbery. Rooster Cogburn, Mattie Ross, and La Boeuf stake out the dugout and smoke them out leading to Moon and Quincy getting shot by Rooster. Rooster then questions them. Quincy won't talk and threatens Moon with a knife. Moon finally talks and Quincy knifes him. Rooster draws his gun and shoots Quincy dead for killing a witness. Rooster then hides Quincy's body until Ned is gone then he takes it back to McAlester where Quincy is buried.