Prerequisites: Max out Leano's level 1 friendship
Reward: None
Unlocks: Leano level 2 friendship, A Beloved Sword
First step's first: Become friends with Leano. The process is the same for every character, though you need to find her first. Leano is the owner of the Primerose Sail, a fishing boat in the northwest of Moonbury that also serves as the local fishing shop - and her home. You can find Leano out here most of the time, either fishing, idling at a table near the Primerose Sail, or in her room on the boat. Build a relationship.
Once you max out the first level of your relationship with Leano a message will appear on the Bulletin Board in the Town Square, inviting anyone who wants to fish to join Leano at the Primerose Sail between 9:00 and 16:00, from Tuesday to Sunday. Visit her and Leano will challenge you to catch a bundle of fish. In exchange she'll tell you stories about her supposed history as a pirate.
You need to catch twenty pieces of White Meat to satisfy Leano's appetite for fishing. White Meat can be fished up from (almost) every catch that you'll make at the Park Fishing Spot, in the southwest of Moonbury Town. This will take a little while, but it's not difficult - and if this is your first time fishing, well, you'll get in plenty of practice in the process. Buy more Worms from the Primerose Sail if you run out. They're only 5 G apiece, and won't hurt your wallet too much.
(This quest appears to be a bit bugged at the moment. The original demo required 10 pieces of White Meat and 5 pieces of Pink Meat, and despite the initial description that still seems to be the case. You can get Pink Meat from the Mackerel in the Park Fishing Spot.)
Track down Leano and speak to her. You'll jump over to the Primerose Sail for a quick conversation, and your friendship with her will be boosted to the second level. You'll also pave the way to the next relationship quest, A Beloved Sword.