Can I complete a quest during the "Waterdeep Harbor" phase?
Yes. But if you have enough cubes to complete that many missions, you'll probably find the spaces at Waterdeep Harbor filled with enemies playing mandatory quests on you...

From the FAQ:


  Q: When you move (reassign) an agent out of "Waterdeep Harbor" to another building, are you taking another turn? Can you complete a quest?
  A: You can complete a quest after moving (reassigning) an agent out of "Waterdeep Harbor" - it is another turn.