He is a Swedish international. Born in Stockholm to a Ghanaian father and a Swedish mother, Quaison started his professional career in 2011 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 Mjallby.
He scored his first goal on 20 May the same year, in a 5–2 victory against IFK Norrköping. He made 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.
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 23 January 2013, Quaison made his debut for the Swedish national football team, against North of Korea in the 2013 King"s Cup. Three days later he scored his first goal for Sweden in a 3–0 victory against Finland in the final of the tournament.