I'm aware that inductive loads like relays need a flyback diode. Since a motor is also an inductive load, I'm inclined to put a reverse-biased 1N400x diode across the terminals of this fan. However, I've never heard any mention of using flyback diodes with fans. Am I correct in thinking I need a diode, or is there some reason why I wouldn't? Thanks!
It's a brushless DC motor. Most, if not all of them, have flyback diodes across the windings (inductors) which are switched electronically with transistors in the motor (contrast to mechanical commutators in brushed motors).
There shouldn't be any need for a flyback diode across this fan, unless there is some significant series inductance eg a EMI filter inside the motor.
But, when in doubt, add a flyback diode. It's cheaper to add one unnecessary diode than frying your circuit because you did need one.