Heaps began his freshman season sharing time with starting quarterback Riley Nelson. Once Nelson was lost to injury against Florida State, Heaps led the Cougars the rest of the season to six wins and four losses. He improved as the season progressed and had only a single loss in his last six games as he surpassed Ty Detmer's freshman touchdown record that stood for 22 years with 15 touchdowns. In the month of November and the December bowl game, Heaps threw 13 touchdowns and two interceptions and passed for 1,259 yards. He was the first freshman quarterback to ever lead the Cougars in Brigham Young's 29 bowl game appearances. Heaps was named the New Mexico Bowl's Most Valuable Player, breaking some of the bowl's records after leading BYU to a comfortable victory with a quarterback rating of 122.5 throwing 4 TD's with 1 INT. As a true freshman, Heaps threw for 2,316 yards, 15 touchdowns, and 9 interceptions with a quarterback rating of 116.2. He completed 219 passes on 383 attempts with a completion percentage of 57.2%.
During the 2011 season, Heaps started the first five games of the season. During those first five games, Heaps threw 3 touchdowns and 5 interceptions and failed to garner a QB rating higher than 116.3 or complete a pass greater than 37 yards. He was benched during the Utah St. game in which backup quarter-back Riley Nelson led BYU to a come-from-behind victory. Heaps saw limited action until Riley Nelson was injured during a game against the University of Idaho. Heaps finished the game against Idaho, and started the next game against New Mexico State, but was replaced when Nelson was cleared to play the following week. Heaps ended the 2011 season with a completion percentage of 57.1, 9 touchdowns, 8 interceptions, and a QB rating of 111.0. After the December 3 game against Hawaii, Heaps announced that he would not travel with the team to the Armed Forces Bowl game, but would instead be seeking to transfer schools.
Following BYU's 2011 season, Heaps announced he would not participate in the 2011 Armed Forces Bowl game and that he would be transferring to the University of Kansas to play for new head coach Charlie Weis. Due to NCAA transfer rules, Heaps redshirted for the 2012 season. He was expected to compete for the starting quarterback position at Kansas with Michael Cummings. On June 13, 2014, the University of Kansas announced Heaps was leaving the program for another school.