Not to exclude Shieldy, of course, they're best friends.
Episodes Suggesting a Friendship
In How It All Began, they spoke at the very beginning, and agreed to team with Shieldy during the challenge.
In BOTO: The Videogame, Boombox was the first to be chosen by Party Hat.
In Unprepared To Be Scared, Party Hat moans the loss of Boombox.
Episodes Suggesting a Conflict
There are none so far. How nice.
Both were recommended characters in other shows.
Party Hat shows up in II Ep. 14, because Anko6 won Adam's fan art contest, and he could pick anyone to be in the episode. He demands a boomerang, and when it fails to go to him he cries.