"The Quest" means the Hidden Object puzzles you have to solve to open the books.
If you could de-activate the fairy, the first one would obviously be the easiest and, more importantly, the easiest to re-start. Unfortunately, the game saves each object part you find instantly. And since the fairy talks too slow at the beginning, you have to do this achievement on one of the later three.
It's best to backup your savegame file after you've done all 16 levels of a book plus the four Spot the Difference puzzles. It's an .XML file located [under Vista/7] at C:\ProgramData\Playrix Entertainment [or use Start | Run | %AllUsersProfile%\Playrix Entertainment].
The achievement should unlock the moment the key appears [i.e. you used the last of the four items with the last "combinable" area of the puzzle].