Good question that is. I think I did nothing with the governor concerning pirate quests.
As mentioned, I returned from Barbados, where I had plundered lots of ships. When I reached Martinique, I had 3 prizes with me. Talkked to governor first, saw the promotion dialogue option, got promoted, even twice (yeah, it were a lot of ships I plundered ...). I did not ask him about pirate quest at that time.
I sold my booty, sold the prizes, repaired my ship, baught equipment for my crew, hired 3 new officers ... and then talked again to governor. Wanted a pirate hunt quest then, but he did not have one. Waited 3 days, asked again, same result. Sailed to Guadeloupe, to try other governor, same result.
No idea why that happened.
I'll take my old Martinique game just after arrival there and try again,a nd see what happens.
Thanks for looking in that.