Quinn joins the Jets having been released by the Seattle Seahawks during the final roster cuts on Saturday, missing out to Tarvaris Jackson in the race to become back-up quarterback to Russell Wilson.
The 28-year-old was delighted to have joined the Jets, who open their regular season against the Tampa Bay Buccaneers on Sunday.
"It's a great opportunity. The Jets are a historic organization," Quinn told reporters.
The Jets released quarterback Graham Harrell less than a week after signing him in order to make room on the roster for Quinn.