Dog in Dredge

On the outside of one of the islands in Star Bay, you will hear barking. Swim closer (this is to the left of the place with numerous houses) and go ashore. Call the dog, give him the fish and stroke it. Take your dog with you.

You won't have to search for a long time for the owner - you can give the dog to the Scientist from the Old Fort. And you can leave the dog after talking with her. The task will end anyway. The dog will spit out the treasure anyway. That is, leaving it at home does not make sense - the animal does not give any bonuses.