Well, it would be really wild. Naruto's energetic and positive nature might clash with Beerus' somewhat lazy yet powerful demeanor. Naruto could learn some unique combat skills from Beerus. Maybe he'd be able to use godly ki in addition to his chakra. And Beerus might be annoyed by Naruto's constant need to prove himself at first but gradually grow fond of him.
Naruto's character would change significantly. He'd likely be more self - assured as he grows up under Beerus. He may not have the same sense of community as he did in Konoha since Beerus is more of an individualistic being. He might also be more patient and less likely to jump into action without thinking, having learned from Beerus' more calculated ways.
Well, one possible plot could be that Naruto is somehow brought to the Dragon Ball universe and is raised by Beerus. Beerus, being a powerful god, could train Naruto in ways that combine the techniques of the Naruto world with the power levels of the Dragon Ball world. Maybe Naruto would learn to use chakra in new ways influenced by ki. And together, they might face threats that are common to both universes, like some powerful villains who want to take over both the Naruto world and the Dragon Ball world.
There are so many fanfics out there that it's difficult to point out just one 'dbz Beerus Naruto raised' fanfic. However, when looking for such a fanfic, consider using specific tags while searching. For example, tags related to the relationship between the characters (if any in the fanfic), the setting, or the overall theme. Maybe a tag like 'crossover' along with the names of the characters. This way, you can narrow down your search and find something that might interest you.
Beerus would likely be more powerful. He is a god of destruction in his own universe with incredible cosmic powers. Naruto is extremely powerful in his world, but his power is mostly based on chakra and ninjutsu, which is on a much smaller scale compared to Beerus' ability to destroy planets with a flick of his finger.
The plot might center around Goku being raised by Beerus in a more secluded place. Beerus could be training Goku to be his successor as a god of destruction. Goku would have to learn about the balance of destruction and creation, which is different from his usual Saiyan power - up goals. This would also mean that Goku would have a different view on the universe and its inhabitants. He might be more detached in some ways, but also more powerful and with a different moral code influenced by Beerus.
Well, Naruto could be in awe of Beerus' power. Maybe he tries to learn some of Beerus' combat techniques. Since Naruto is known for his determination, he might pester Beerus to train him.
His character would develop a more strategic side. Beerus is a very strategic fighter, and Goku, being raised by him, would learn to think more before acting in battles. He would also become more aware of the power hierarchy in the universe at a younger age, which could make him more cautious or more ambitious, depending on how you look at it. Additionally, his Saiyan pride might be tempered by Beerus' teachings, making him more of a team player in some ways.
It could be someone who wants to disrupt the relationship between Naruto and Beerus. Maybe a powerful dark ninja who sees Naruto's newfound connection with Beerus as a threat to his own plans for power.