He played college basketball for the Kentucky Wildcats and was selected by the New York Knicks in the first round of the 2020 NBA draft.

While attending The John Carroll School, Quickley had a breakout sophomore campaign and averaged 17.7 points, 3.9 rebounds, 2.9 assists and 1.5 steals per game.

[3] He posted 20.8 points, 6.7 rebounds, 6.7 assists and 3.7 steals per game as a senior and led the team to the Maryland Interscholastic Athletic Association A Conference title.

[21] On December 30, 2023, the Knicks traded Quickley, along with RJ Barrett and a 2024 second-round draft pick, to the Toronto Raptors in exchange for O.G.

[24][25] On April 5, 2024, Quickley helped the Raptors break their 15-game losing streak, scoring 25 points in a 117–111 win over the Milwaukee Bucks.

[27][28] Quickley played for the U.S. national under-19 team in the 2017 FIBA Under-19 Basketball World Cup, where he was coached by John Calipari.