Tarik Coleman rushed for 163 yards and a TD, leaving him 88 yards short of becoming the Aggies all-time leading rusher with three games left.
Quick scored from the 6 in the first quarter and hit Caleb Gabriel on a 26-yard pass play as NC A&T built a 28-0 lead early in the third.
Florida A&M (1-7, 1-4) got on the scoreboard late in the third quarter when Zachery Webster caught a 2-yard TD pass. The Rattlers added a field goal in the fourth.
North Carolina A&T is No. 20 in the FCS polls, their highest ranking since 2003.