Yes, you can continue your main scenario quests with your Dark Knight.

As long as your current level on any class/job you are at the moment is at or above the level of a quest you can accept it and fullfil it.
Some quests have special conditions to accept them but they are shown in the acception screen if you don't meet them.
Also some quests can only be finished with the same job as you accepted them, but this is rare and not in the MSQ.