Quinn was starting in place of Matt Cassel, who was benched after a weak start to the year but entered the game when Quinn got hurt.
Quinn was hit hard on a sack by Rolando McClain on the Chiefs' third possession, and then took another shot when he threw an interception off his back foot.
Cassel came into the game for the Chiefs' next series.
Cassel sustained a concussion three weeks ago against Baltimore and missed a game at Tampa Bay before getting benched. He was cleared to practice during the Chiefs' bye last week.
Running back Nate Eachus left later in the game with a head injury.