Quaison started his professional career in 2011 on loan at
Väsby United.
AIK Quaison joined
AIK in 1997. He made his first match for AIK on 1 April 2012, as a substitute in a 0–0 draw against
Mjällby. He scored his first goal on 20 May the same year, in a 5–2 victory against
IFK Norrköping. His second goal came in the 3–1 win against
BK Häcken on 8 July 2012. He made
UEFA Europa League his debut in a 4–0 loss to
Napoli on 20 September 2012 before going on to playing a further five times in the Europa League cup run. He received the first red card of his career against
Halmstads BK in a 3–3 draw. He finished his second season making 28 appearances in all competitions. On 6 August 2013, Quaison scored a long-distance goal against
Manchester United in a 1–1 draw during a pre-season friendly.
Palermo In July 2014 Quaison moved to the
Serie A club
Palermo, signing a three-year contract, having been previously linked with
Stoke City,
Leeds and
QPR. On 17 December 2019, Quaison scored his first career hat-trick in a 5–0
Bundesliga victory over
Werder Bremen. On 16 May 2021, Quaison scored the sole Mainz goal, a second-half stoppage time penalty, in a 3–1
Bundesliga loss to
Borussia Dortmund, for his thirtieth goal for the club. In doing so, he became the sole all-time top Bundesliga goalscorer for Mainz, breaking a tie with
Yunus Mallı and
Mohamed Zidan.
Al Ettifaq In July 2021, Quaison signed a contract with
Saudi Pro League club
Al Ettifaq. On 21 August 2021, Quaison scored his first goal for his new club in a 3–3 draw against
Al-Shabab.
Aris On 23 September 2024, Quaison signed a two year contract with the Greek
Super League 1 club,
Aris Thessaloniki. He played his first match for the club on 5 October 2024 when he came in from the bench in the 87th minute in a 2-0 win over
Lamia. On 22 April 2025 and again on 22 August 2025, Swedish media reported that the club wanted to cancel Quaison’s contract. In both cases they cited Greek newspaper Gazzetta as a source. On neither of these occasions was an official statement made by the club or player. On 22 December 2025, Swedish newspaper
Expressen reported that Quaison and his former club, AIK had been in touch with each other about a possible return to Sweden.
Gulf United On 6 February 2026, Quaison signed a contract with
UAE First Division League club
Gulf United. ==International career==