I think it depends on various factors. Goku's determination and growth are remarkable, but Beerus has an overwhelming level of power and authority that might be difficult to overcome.
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.
Well, in many goku x beerus fanfiction, the theme of cultural exchange is quite common. Goku represents the Saiyan culture and Beerus the divine culture of the Gods of Destruction. Their exchanges can be really fascinating. Also, the theme of redemption can be found. For example, if Beerus has made some mistakes in the past, through his relationship with Goku, he might seek redemption. And of course, the theme of facing a common enemy together is prevalent. They combine their powers and strategies to take on threats that neither could face alone.
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.
One common theme is power growth. Goku is always seeking to get stronger, and in fanfictions with Female Beerus, she often challenges him in new ways, pushing his limits and leading to his power growth.
One popular goku x beerus fanfiction is 'The Unlikely Bond'. It explores a deeper friendship between them that goes beyond their battles. Another is 'A New Power', where they team up against a new and powerful enemy. There's also 'Shared Training', which focuses on them training together and how their relationship evolves during that process.
One scenario could be that Goku surprises Beerus with his speed and initial attacks as SSJ4. But then Beerus counters with his hakai and forces Goku to dodge constantly. Another could be that Beerus underestimates Goku at first and Goku manages to land some powerful blows.
Well, in many fanfictions, SSJ4 Goku is often depicted as having a chance against Beerus. SSJ4 Goku has shown incredible strength and combat skills. But Beerus has the power of a God of Destruction. His hakai ability could potentially wipe Goku out. Still, if the fanfiction is written in a way that gives Goku some new power - up or a special advantage, he could win. It really depends on the creative direction of the fanfic writer.
One popular Goku and Female Beerus fanfiction might be 'The New Bond'. It focuses on their unique relationship development as they face new threats together. In this story, Goku's determination and Beerus' power combine in interesting ways.