These quests can be obtained from message boards on your ship and in towns, under the name of the quest it will tell you if it's a Bounty, War Contract, or Naval Quest, you will also be able to see which they are in your quest log. Bounties will have you assassinate someone or wipe out an encampment. War Contracts will task you with destroying x amount of nation supplies, killing x amount of soldier types, etc. Naval Quests will have you doing something ship based.
War Contracts are something you should pick up even after you've gotten this achievement, the other two can be grindy but contracts are things you will likely do just by playing the game naturally.
Edit: ImaginaryRuins said they received their achievement after completing a naval conquest battle, these may also count as a "naval quest".
Edit #2: You can keep track of the number of Bounties and Contracts you have completed by looking under head engravings and checking the number under the engraving for + Ranged Resistance. The first tier is 5 and the second 10. So that gives you 15 and you just need 5 more from the 15 tier 3 requires. Sea Bounties don't count for this, so it is only really useful if you only do land things, credit to Seeker74.