Several of Colquitt’s family members have punted in the NFL. His father, Craig, won two Super Bowl rings playing for the Pittsburgh Steelers. His older brother, Dustin, spent 15 seasons as the punter for the Kansas City Chiefs and won a Super Bowl ring after the Chief’s victory in Super Bowl LIV. His cousin Jimmy punted at Tennessee (1981–1984); his 43.9 punting average ranks first in school history. Jimmy played in the NFL with the Seattle Seahawks in 1985.

While at the University of Tennessee, Colquitt had a well-publicized saga with alcohol problems. He was first suspended from the University of Tennessee football team in March 2004 after three alcohol-related arrests, despite having only arrived on campus in 2003. In his junior season, he was suspended for the first five games of the season and had his scholarship stripped after being arrested on charges of DUI and leaving the scene of an accident.

Colquitt played college football for the Tennessee Volunteers at the University of Tennessee, where he majored in political science. During the 2005, 2006, and 2007 seasons, he was the Volunteers first team punter. He earned consensus first-team All-SEC honors in 2006 and second-team All-SEC honors in 2007.

Colquitt was signed to the practice squad of the Miami Dolphins on December 22, 2009.

Colquitt was signed to the Denver Broncos’ active roster from the Miami Dolphins’ practice squad on December 30, 2009.

On September 12, 2010, in the season opener against the Jacksonville Jaguars, Colquitt made his NFL debut and had four punts for 172 net yards (43.00 average).

In the 2011 season, Colquitt accumulated 4,783 yards, enough for fifth all time for single-season punting yards in a season.

On April 23, 2013, Colquitt signed his restricted free agent tender. On August 11, 2013, he signed a three-year extension worth $11.6 million on top of his one-year tender. The Broncos reached Super Bowl XLVIII, which they lost to the Seattle Seahawks by a score of 43–8. In the loss, he had two punts for 60 net yards.

Colquitt agreed to a $1.4 million pay cut on August 3, 2015.

On February 7, 2016, Colquitt was part of the Broncos team that won Super Bowl 50. In the game, the Broncos defeated the Carolina Panthers by a score of 24–10, giving Colquitt his first Super Bowl victory.

On August 30, 2016, Colquitt was released by the Broncos due to not accepting a pay cut.

On September 3, 2016, Colquitt signed a one-year deal with the Cleveland Browns worth $1.7 million.

On February 28, 2017, Colquitt signed a four-year contract extension with the Browns.

On August 31, 2019, Colquitt was released by the Browns.

On September 1, 2019, Colquitt was signed by the Minnesota Vikings.

On March 19, 2020, Colquitt signed a three-year, $9 million contract extension with the Vikings.