This is odd. The target for any governor ship-hunting quest is "Quest Pirate", who is given a new name, ship and nation by the quest and then set to appear somewhere near the island when you set sail. In your savegame at sea, he's no longer near the island - he's nowhere at all. I've no idea how that could have happened as "Quest Pirate" does not seem to be used in any other quest, and you can't take a second ship-hunting quest from another govenor until you've completed this one.
There's also a savegame with you standing in front of the governor. Using that, I was able to take a ship-hunting quest and complete it without any trouble, mainly because I used cheat mode to make myself indestructible so I could capture the ship easily. The point is, the ship did appear, the quest did complete, and it also turned me into a pirate.
This is because you're flying a French flag. America is at war with Britain. Probably due to a random relation change which can happen in FreePlay, France is at peace with Britain. The quest ship is automatically hostile anyway. So, flying a flag which is not hostile to Britain, I captured a British ship, and that's about the quickest way in the game to turn pirate!
I tried again - loaded the savegame in front of the governor, took a ship-hunting mission, hoisted an American flag as soon as I put to sea, and then went after the target. This time I completed the mission and did not turn pirate.
And you'll want to use the American flag if you want to continue attacking British ships on behalf of the US Navy.