It's a bit hard to say for sure. Both Vegeta and Goku have their moments of strength. Goku has some unique powers and techniques that give him an edge at times, but Vegeta is no slouch either and constantly works to catch up and surpass Goku.
His power would develop more in line with Saiyan traditions. He'd probably have a stronger base power due to the intense training environment on Planet Vegeta. There would be more emphasis on raw power and combat skills from a young age.
Well, he would probably be a very different person. He might be more in tune with the Saiyan warrior culture. He could be more aggressive and less empathetic compared to the Goku we know. He would grow up learning the ways of the Saiyans on Planet Vegeta, which means he'd be trained to fight and conquer from a very young age.
In this fanfiction, Goku has the potential to be stronger. He has mastered different forms like Super Saiyan and beyond. However, King Vegeta is not to be underestimated. He has his own Saiyan pride and power. If it comes down to a fight, Goku's adaptability and continuous growth in power might give him an edge, but King Vegeta's experience and strategic mind could also make it a very close battle.
In fanfiction, it's possible for Naruto to be stronger than Goku. Fanfic writers often tweak the power levels and rules of the characters' universes. For example, if the fanfic writer decides to nerf Goku's Saiyan abilities while enhancing Naruto's Sage Mode and all his ninja skills to an extreme level, Naruto could end up being stronger.
In some fanfics, Naruto can be made stronger than Goku. Fanfic writers have the freedom to manipulate the power levels and abilities of characters. They can give Naruto new and overpowered abilities that surpass Goku's. However, in the original canon of their respective series, Goku has extremely high power levels and abilities that are on a different scale.
In some fanfictions, it might be a really strange and unique concept where Goku is depicted as Vegeta's pet. But this is far from the original relationship in the official Dragon Ball story. It's just a creative and often humorous take that some fanfic writers explore.