Quinn was born in Buffalo, New York and he graduated from Siena College. He was an English teacher in the Orchard Park Central School District and served on the Hamburg, New York Town Council from 1982 to 1984, followed by a stint as town supervisor until 1993. After the retirement of Congressman Henry Nowak, Quinn was elected to the House of Representatives in 1992 representing most of Buffalo and suburban Erie County. He defeated Democratic Erie County Executive Dennis Gorski, in a surprising upset. He was reelected five times, by surprisingly large margins for such a heavily Democratic district.
Quinn had close ties to organized labor, something not common with most Republicans. Despite a close relationship with Bill Clinton, he voted for three of the articles of impeachment against him. There have been several suggestions that the House leadership pressured him into doing so, which were personally denied by Quinn in a 1999 article in the Washington Post. It has been reported that Clinton's visit to Buffalo after his 1999 State of the Union Address was in response to Quinn's vote.
During his tenure in Congress, Quinn was an active member of the Transportation and Infrastructure Committee, where he served as chairman of the Railroads Subcommittee.
During the 2002 redistricting process, one map had Quinn drawn into a district with a longtime friend, Democrat John LaFalce. However, he ended up with a district that encompassed many parts of his old district and added Chautauqua County. The new district was only slightly less Democratic than his previous district; Al Gore would have defeated George W. Bush by a sizable margin under the new district boundaries. On April 26, 2004, Quinn announced he wouldn't seek re-election. After leaving Congress, Quinn joined the lobbying firm Cassidy & Associates. He was appointed President of Erie Community College in 2008