Are there any quests that involve ships?
Also are there any islands to go to in this game?

Yes and Yes. The main plot requires you to take a ship to an island. You can also take a ship back from this island.

You can't choose to go to other islands and you can't hire a ship yourself. The ship sequence is linear (you are stuck on one path).

Also, the graphics only show a map like you are looking at a ship's deck. There is really nothing to explore on the ship.