No, you don't need to do the whole quest in order to be able to do the daily task lucky charms everyday.
On the first time you'll do the mission, ask for mission to Rottin Wood. The first one will already be the lucky charms mission. After your finished the mission, instead of mission you'll start saying lucky charms everyday.
Please notice that you can even do the daily task on the same day you did the mission, thus doing it twice on the first day ever.