Boar Hunt!

"One of the villagers is having problems with a wild boar.
Those boars can be a real handful. Maybe you could help out?"

"Boar Hunt!" is a quest in NieR and NieR Replicant ver.1.22474487139.... Completing this quest unlocks the ability to ride on boars, as well as the "Call Animal" feature on the menu.

Activation

An old man sitting next to the item shop in the shopping district says that a wild boar has been injuring people in the northern plains. It's time to end its rampage.
You defeated the wild boar. Report back to the client.
The client said that a boar's tusk will allow you to tame and ride the animals.

This quest, alongside the "Fisherman's Gambit" chain, are the only quests that carry through to Part 2.
The accompanying dialogue for this quest will sometimes trigger once you leave The Aerie with Kainé agreeing to join the party, even if the quest hasn't been acquired.
A good strategy for defeating the boar is to hide behind a rock and let it charge into it. When it's stunned, hit it with Down Attacks (Guard+Attack or the Action button in ver.1.22...) until it gets up, then hide behind the rock and repeat.

This quest is referred to as "A Sheepish Request" in Grimoire NieR: Revised Edition.^[1]