Connette's mother, Nancy, was diagnosed with cancer in 2013. The Blue Devils have fulfilled Connette's request to provide permission to schools close to his home of Corona, Calif.
He will have immediate eligibility at his new school because he graduated with a degree in economics from Duke in December.
"First and foremost, our prayers and concerns are with Nancy Connette and her quest for good health," Cutcliffe said in a statement. "We thank Brandon for his many contributions to Duke University and our football program. We want nothing but the best for Brandon -- both on the football field and off -- and his personal situation is very close to our hearts. He's been an integral part of our program over the past four years, and we wish him continued success going forward."
Connette, who split time with Anthony Boone last season, appeared in 13 games and started three, helping the Blue Devils go 10-4 and win the ACC's Coastal Division.
He completed better than 62 percent of his passes for 1,212 yards with 13 touchdowns and six interceptions. He added 337 rushing yards and 14 more touchdowns on the ground, and his 27 total touchdowns set a school record.
Connette is the school's all-time leader with 31 rushing touchdowns.
Boone is now the clear No. 1 at quarterback for Duke as it enters the 2014 season.
"I could not be more appreciative of Coach Cutcliffe for his understanding of this situation," Connette said in a statement. "Being away from my family during my mother's illness has been the most difficult thing I've ever had to endure. I'm thankful that Duke University afforded me so many incredible opportunities -- academically, athletically and socially -- and will always hold my experience here in the highest regard."