Are you sure you haven't already completed the quests prior to the release of 2.0, and the Taken King? Personally, I barely touched my Titan and Warlock. That's why when I signed into Destiny for the first time after 2.0 launched, I found all sorts of missions and quests that weren't completed prior to Destiny's revamp. On my Hunter, since I already completed The Dark Below and House of Wolves quest lines, they weren't available for that specific character. To answer your question directly, quests aren't shared between characters, so you can't transfer them. Any quests you failed to complete prior to 2.0 (regarding TDB & HoW) should have been available in your Abandoned Quests kiosk.