Can Trophies Be Unlocked In Battle Royale Mode?

There aren't really any trophies that can be unlocked. Most of the existing trophies are geared toward completing missions in the save the world game mode. Even the kill trophies are related to killing husks and not other players, so even if it was possible, there aren't any trophies in the Fortnite trophy list that can be shared between the 2 game modes. I would expect the game to get an update in the future that adds some trophies for the Battle Royale game mode once it is out of beta and has more content to add trophies for.