Here's an example from my actual play:
One player had a character with the "merciless" trait. As GM, I decided to poke at that trait: I sent a group of peasants to the gate of the castle that was under siege and had them demand to be let in. As captain of the guard, he had the authority to decide what was to happen to them. He had a Belief about finding food for the castle's current inhabitants and knew full well that there wasn't room for more mouths to feed. Would be be merciless and turn them away? I think I explicitly pointed out that he had the trait.
In the end, he did, but explained that he wasn't entirely being merciless, since he felt they'd have a better chance in the forest to the east than if they were penned up in the castle with everyone else.
Later in the game, there was a situation where the players were trying to decide whether they should kill a man who'd seen something they'd done that they wanted to keep secret. I goaded the merciless character's player: "Well, are you merciless? Will you kill this man to keep the secret? ...this poor, innocent man who just happened to be in the wrong place at the wrong time? Or will you spare his life, even though he'll almost certainly reveal what you did, with all the severe issues of diplomacy that would cause between you and the dwarves." It definitely caused some internal struggle - if I recall correctly, he earned Moldbreaker that session.
Beliefs should generally take precedence, but character traits are another good source of compelling situations to put your players in, either by themselves or as flavoring for the Belief challenges. Don't leave it up to the players' initiative, find out for yourself what those adjectives written on their sheet really mean, and just how far they go!
(If your players are like my players, and tend to forget they have traits most of the time, my experience is that there is nothing wrong with giving them some explicit reminders until they start to get the hang of it themselves, or to be a bit ham-handed at first. Don't give them their Artha for them or force them to do anything, but don't shy away from leading them right up to the brink and gesturing suggestively forward. If they repeatedly decline to play the trait, that's fine - it'll be reflected in the trait vote, most likely, but better to vote a trait off because the player actively rejected it in play than because everyone kept forgetting it existed and so it lay fallow for the whole adventure.)