Here's a screenshot of me talking to the governor of Cayman, Edmund Christobel Shaw:
He's supposed to lead with the start line for the quest to find his daughter, which would be "Well, I have a mission if you are interested." Instead he leads with the generic governor line about hostiles. You, however, have the option to ask about the quest, using the line "Maybe. What is it exactly?" which would be your reply to his line about a mission if you are interested.
I would guess that any other governor who has a specific lead-in line to a quest won't use it either, but will always lead with the generic line about hostiles and possibly the bit about smugglers. (The latter appears to be conditional on you being in his nation's service, which in the above case I'm not.)

He's supposed to lead with the start line for the quest to find his daughter, which would be "Well, I have a mission if you are interested." Instead he leads with the generic governor line about hostiles. You, however, have the option to ask about the quest, using the line "Maybe. What is it exactly?" which would be your reply to his line about a mission if you are interested.
I would guess that any other governor who has a specific lead-in line to a quest won't use it either, but will always lead with the generic line about hostiles and possibly the bit about smugglers. (The latter appears to be conditional on you being in his nation's service, which in the above case I'm not.)