Prerequisites: Max out Matheo's level 2 friendship
Reward: None
Unlocks: Unlock Matheo level 3 friendship, A Mysterious Disease
Once you've completed the first friendship quest with Matheo you'll need to go back to befriending him. As before, you'll need to spend a lot of time visiting Matheo's home and bringing him presents. If you haven't been doing so already, you may want to befriend Matheo, Bubble, and Forrest all at the same time, as you'll likely see Bubble and / or Forrest on your way to Matheo's house. Might as well save time.
Max out the second level of friendship with Matheo - against his wishes - and a notice will go up on the Bulletin Boards of Moonbury Town, noting that Matheo is behind on some work. Visit Matheo between 8:00 and 19:00, Monday to Sunday, and Matheo will confess that he has... again... lost his notes. You'll need to hunt through Meadow Range to find Matheo's Report so he doesn't get in trouble.
Despite what Matheo says, he absolutely did not lose his papers in Meadow Range. To find them you'll need access to Glaze Iceberg, the second area you can unlock. Make your way to the east side of Glaze Iceberg and check the above above the North of Glaze Iceberg Fast Travel point. There's a clearing here where you'll find Matheo's Report, scattered across the snow.
Take the notes back to Matheo and he'll be... slightly... more grateful than last time. He's still pretty huffy about accepting your help, though, and quickly shoos you away. Ah well. The quest will end here, and you'll move up to the third level of friendship with Matheo. Continue to befriend him and you'll unlock the third quest in Matheo's storyline, A Mysterious Disease.