The Mer-ky Depths side quest in Hogwarts Legacy becomes available during the Prepare for Your Search for the Final Keeper main mission. Travel to the Boathouse Floo Flame at The Great Hall and speak with Nerida Roberts to learn about her desire to become a liaison between the wizarding community and the Merpeople. Afterwards, swim out west of where Nerida is sitting to find a whirlpool then dive down into it to reach an underwater cavern.
There will be 3 moths here you need to lead to the main door to proceed further. Two of the moths are stuck behind a spider web. Burn the web away using a fire spell then use Lumos to attract the moths one at a time and lead them to the door.
This will open it revealing a chest. Loot it to receive A Mermish Artefact. Take the item back to Nerida. You can choose to either give it to her or not. Either way you’ll be rewarded with the Mermish Liaison Uniform appearance and the quest will complete.