This is part of a first-playthrough guide using the Faerie story arc. It covers both the original PSX version and the 2021 HD remaster, though some place and item names differ slightly in the latter. You can find an overview of the Faerie% path here.
The Quick Version
Return to Gato Grottoes and find Niccolo in the item shop. Agree to accompany him to the Waterfall to collect Greenballs. Wander about a bit as you wait for him, and eventually you’ll encounter a faerie. The quest ends shortly thereafter, and you get the Broken Doll Artifact in exchange for about two minutes of your time.
The Details (May Contain Spoilers)
This quest is super-short, but the exact things you need to do to progress the quest state are pretty obscure. After you meet Niccolo in the item shop and agree to help him, head for the lower-left (cave) section of the Gato Grottoes map and choose “To the Waterfall” on the fast-travel menu. Go up to the next map node and you’ll be at the Waterfall.
Once there, head to the right (Pathways of Ascetism), then immediately return to the Waterfall and wait by the log bridge until the faerie appears. Talk to the faerie and answer its question, then answer Niccolo (first dialogue option is fine in both cases). Niccolo will announce that he’s finished collecting Greenballs and award you the Broken Doll.