McKnight was born and raised in Bridgeport, Connecticut. He is the son of Vanessa Bruce and Terry McKnight. His mother died of cancer when he was 13 years old, shortly after he won a diocesan championship. Quincy used her death as motivation to become better at basketball going forward. McKnight began his high school career at St. Joseph High School in Trumbull, Connecticut and led the team to back-to-back state championships. As a junior, he averaged 22 points per game but injured his knee in the spring. He transferred to The Phelps School, where he competed in basketball and tennis. McKnight was an honor roll performer at The Phelps School and was coached in basketball by Brian Shanahan. In AAU play, he competed for the PSA Cardinals in the Peach Jam. As a senior at The Phelps School, he averaged 15.9 points, 4.2 rebounds, and 3.5 assists per game and was selected to play in the 2015 Jordan Brand Classic Regional Game. He led the team to the 2015 PAISSA state championship. McKnight was lightly recruited, being regarded as a two-star recruit ranked no. 341 in his class by 247 Sports and signed with Sacred Heart.
College career
As a freshman, McKnight averaged 11.4 points and 5.3 rebounds per game. He was a five-time Northeast Conference rookie of the week honoree and was named to the league All-Rookie team. McKnight scored a career-high 44 points in a triple-overtime win against Bryant during his sophomore season. He also posted 36 points in a loss to Boston College and scored at least 25 points in 10 games. As a sophomore, McKnight averaged 18.9 points, 4.9 rebounds and 3.0 assists per game. He was named to the First Team All-Northeast Conference. After the season, he announced his intention to transfer. McKnight scored 914 points in two seasons at Sacred Heart. After impressing in an open gym workout, McKnight transferred to Seton Hall and was offered a scholarship on the spot. McKnight redshirted the 2017–18 season, during which he engaged in fierce practices against Khadeen Carrington. McKnight forged a special relationship with Myles Powell during a summer class in 2017. During his junior season, McKnight emerged as a lockdown defender and secondary scorer to Powell. He averaged 9.4 points, 3.9 assists, and 1.6 steals per game. On December 19, 2019, McKnight scored 17 points and had eight rebounds and six assists, including two free throws with 1.1 seconds on the clock, in an upset win against Maryland without the injured Powell and Sandro Mamukelashvili. The following game, McKnight scored a season-high 25 points in a win against Prairie View A&M. On February 3, 2020, he scored 15 points in a 74–62 loss to Xavier but injured his knee late and was listed as day-to-day. An MRI revealed a muscle sprain, and he returned for the following game against Georgetown. As a senior, McKnight averaged 11.9 points, 5.4 assists, and 1.5 steals per game. He led the Big East Conference in assists and was named Honorable Mention All-Big East after leading the team to a share of the league championship.
