Even on the largest ships, the captain’s quarters would never be called “spacious.” For this reason, a captain’s bed was a small affair. For your modern version, a twin size mattress is a perfect fit. You will want the bed to take up as little floorspace as possible.
If you want to make your captain’s bed feel authentic, a twin size mattress has another benefit. Much like a modern daybed, the captain’s bed had to serve as a seating area; a twin size bed suits this purpose much more easily than a larger mattress would. You might want to give your twin captain’s bed a slightly higher footboard (to match the headboard) and add a back side to serve as a backrest for some cushions and extra pillows. This way, your bed will almost feel like an seating alcove--very cozy.