Belamy wants to make an assessment of how friendly your pokémon are with you.

The Request

Talk to Bellamy in Jubilife Village to accept his request.

Show a Pokémon with High Friendship

You can show Bellamy any pokémon from your party or from the Pastures. If the pokémon that you show him has high enough friendship (which you can increase by walking around with the pokémon in your party, having the pokémon participate in battles, etc.) you will complete the request.

Reward

Afterward, you can talk to Bellamy any time to find out how close you and your pokémon are. In addition, you get 1 Rare Candy.