Note that it's only available in the post game.
The first way to find the Friend Bow will be at Mt. Faraway. After getting a quest to look for a "Mirage Pokemon" the progression will eventually lead to Mt. Faraway and its 60-floor dungeon. While climbing the floors, players should take the time to fully explore the 30th floor because the Friend Bow could appear anywhere. Note that this is the only way to acquire the Friend Bow without paying in Pokedollars or other consequences.
On the other hand, players can also obtain the Friend Bow from the Joyous Tower which is an optional dungeon with 99 floors requiring a strong team with some Rare Qualities. But the Friend Bow here isn't an random drop at a certain place. Instead, the item can randomly be found for purchase in Kecleon stores while exploring the dungeon, but getting the Friend Bow this way will cost 5000 Pokedollars. Alternatively, those who can't afford the price can attempt to steal the item but only if they are prepared with evolved Pokemon to deal with an overpowered security force that will attempt to take it back. Players should be aware that every new attempt in the Joyous Tower will always restart on the 5th floor if they don't find a Friend Bow to buy or can't fight off an angry horde of Kecleon.
Hope I helped!