It could be Frieza. Since he has a long - standing enmity with the Saiyans. His presence would create a great threat that King Vegeta and Trunks would have to overcome together. King Vegeta has his own history with Frieza, and Trunks knows how powerful Frieza can be in the future.
Another possibility is an original character created for the fanfiction. This antagonist could be a powerful Saiyan who has turned against his own race due to some past grievance. King Vegeta and Trunks would have to deal with this traitor within their own ranks while also trying to figure out what led to his betrayal.
It might be a new character created specifically for the fanfiction. Maybe an alien race that has a vendetta against both the Saiyans and Earthlings and sees the union (even if it's just a meeting) between King Vegeta and Bulma as a sign to start a conflict.
It could be a rogue Saiyan from an alternate timeline who has come to their universe to prove his supremacy. He is jealous of the power that Goten and Future Trunks possess and wants to take it for himself.
The relationship could develop in a more emotional way. Maybe Trunks gets injured badly in a fight, and Vegeta realizes how much he cares for his son. He then becomes more protective and affectionate towards Trunks. On the other hand, Trunks could also start to understand his father better, seeing that Vegeta's tough love is actually a way of showing his love and concern. This new understanding could lead to a more harmonious and loving relationship between them.
In this fanfiction, it could be that King Vegeta mistakes Trunks for a threat at first. But Trunks explains his purpose, which is to protect the Saiyan race. King Vegeta then decides to train Trunks in the traditional Saiyan ways. This training helps Trunks become even stronger and also bridges the gap between their two different generations of Saiyans.
It could be a dark sorcerer who wants to prevent Yona from getting the knowledge from King Hiryuu. This sorcerer might be afraid that Yona, with the king's wisdom, will become too powerful and thwart his evil plans for the kingdom.
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.
One potential theme could be family. Vegeta's relationship with his son is explored, showing how a Saiyan warrior like him can adapt to fatherhood. Another theme might be growth, as Baby Trunks learns from Vegeta and Vegeta also grows as a person through this new connection.
Well, it could be that Vegeta, upon meeting Baby Trunks, realizes the importance of family in a new way. He might start to see the future differently, knowing that he has a son to protect and guide. There could also be some cute moments where Baby Trunks tries to imitate Vegeta's tough guy expressions, which would be quite endearing.
In a 'Vegeta meets Trunks' fanfiction, a great plot could be centered around Vegeta finally opening up to Trunks about his own insecurities. Vegeta has always been this tough exterior, but he might share with Trunks his fears about not being able to protect their family or the Saiyan race. This would be a very emotional moment for both of them. Trunks would see a different side of his father, and Vegeta would be taking a big step in their relationship.
Maybe a rogue ninja who has somehow tapped into dark powers related to Kami's domain. This ninja wants to use these powers to take over the ninja world, and Naruto, with Kami's help, has to stop him. In this fanfiction, the rogue ninja could have some personal vendetta against Kami as well, which adds more depth to the story.