I believe this is Dawn Cook's First Truth as per Trying to remember a book involving a girl from a distrusted race becoming a mage and a dragon.
Alissa doesn't believe in magic. Her father's stories about the Hold, a legendary fortress where human Keepers learn magic from the enigmatic Maters, are just that―stories. But her mother insists that Alissa has inherited her father's magical ability, and so she must go to the Hold―the only place her talents can be trained.
On her way, she crosses paths with Strell, a wandering musician from the plains. And though Alissa is not sure she can trust a plainsman, Strell has something she needs―one of her father's old maps. Travelling together, they can reach the Hold before the snow sets in.
But they don't know that the Hold is nearly empty. Something is very wrong and someone believes that Alissa and Strell knows about a book called First Truth.
I believe that when they arrive, Alissa lets Bailic believe that Strell is the one with the talent. Unfortunately, I don't remember if they do have a plotline where he's found out because he falls asleep, but I do remember there is a scene where he essentially hypnotizes Alissa to try to get the truth from her (it's foiled by the trapped dragon taking over for her) and when she awakes, she sees Strell asleep, and realizes that they don't have long before Bailic realizes that she is indeed the person with the magic.