Quinn was born in Albany, New York, where he was educated at a private school and the Boys' Academy. Early in life, he entered the brewery business with his father and subsequently became a senior member of the firm.
At the outbreak of the American Civil War, he was commissioned as a second lieutenant in Company B, 25th New York Infantry Regiment, which was ordered to the defense of Washington, D.C. in April 1861. He was assigned to duty at Arlington Heights.
Quinn was a member of the common council of Albany 1869-1872. He was elected a member of the New York State Assembly in 1873.
He was elected as a Democrat to the 45th Congress and served from March 4, 1877, until his death.
Quinn died on June 18, 1878. He was buried in St. Agnes Cemetery in Menands, New York.
