You'll need a tamed Cat for this. Stray Cats can be found in Villages wandering around. You'll need a few fish to feed the cat; you have to get close to it and hold the fish in your hand, and the cat will approach you slowly and take the fish.
With a tamed Cat now, all you have to do is sleep in a bed (make sure your Cat isn't sitting down). Once you wake up in the morning, there is a random chance your Cat will bring you a random item, and then the trophy should pop.