Prerequisites: Max out Lucke's level 1 friendship
Reward: None
Unlocks: Lucke level 2 friendship
Your first step is, of course, to max out your initial friendship bar with Lucke. The safest place to look for Lucke is in or around the Farmhouse in the north of Moonbury. Not only is this Lucke's home, he also works the front of the store during the day. If you can't find him there then he may be resting on a bench somewhere in the south of Moonbury Town.
Once you fill the friendship bar to full you'll see a notice from Lucke on the town's Bulletin Boards. It will mention a cape on the northern tip of Moonbury, just beyond the Church. Go here and Lucke will be waiting. He'll grill your character on the Capital, and will be particularly interested in acquiring a book called The Principle of Life.
Head back to your house. To the left of the door is a table with some bags on it. Inspect it and you'll find the book Lucke wanted. Take it to him - again, he really likes spending time around the Farmhouse - and you'll see a cut scene that will end the quest, boosting your relationship with Lucke.