Quinn was born in 1993 in London, England.Raised in South London, he attended the independent school Emanuel School from 2007 to 2012; he was awarded a scholarship as part of their drama scholar program. He subsequently attended drama school at the London Academy of Music and Dramatic Art, graduating in 2015.

Following school, he was cast in the main role of Arthur Havisham in the BBC One television series Dickensian. The show premiered in December 2015. The following years, he acted on the London stage at the National Theatre and Off West End. He starred in the 2017 four-part series Howards End as Leonard Bast, a young bank clerk, opposite Hayley Atwell. That same year, he appeared in the HBO series Game of Thrones in Season 7 as a Stark soldier, Koner.

In 2022, he played Eddie Munson in Season 4 of Stranger Things. He was cast in 2019 with filming taking place in 2021.