The Troll Control side quest can be obtained from Brocburrow in Hogwarts Legacy. Brocburrow is located southeast of Hogwarts. Travel there then locate Alexandra Rickett near a campfire. Speak with her to learn she has been tasked with putting down the pet troll that she’s been training.
Head south of the hamlet to find the troll wandering around in the woods. The troll is level 29 and comes with a giant health bar. Take the beast down the return to Alexandra to inform her the deed has been done. The quest will be completed and you’ll receive a Troll Hat appearance.