Often, Vegeta becomes more in touch with his emotions. He starts to care about things other than just fighting, like his family with Bulma. He may show more love and protectiveness towards her and their child.
In many fanfictions, Vegeta starts as an outsider. He's a Saiyan in a world full of mages. His initial development is about adapting. He has to learn the rules of the Fairy Tail world. For instance, he might not understand the concept of magic guilds at first. As he spends more time there, he becomes more open - minded. He starts to see the value in the friendships and teamwork that are so important in Fairy Tail.
The relationship might develop gradually. Bulma could start by being attracted to Vegeta's power and mystery. As she gets closer to him, she discovers his more complex personality. Vegeta may initially resist her advances but eventually gives in to his own feelings.
In such fanfiction, Vegeta might make Bulma happy by showing his softer side. Maybe he cooks a special meal for her. Vegeta is often portrayed as tough, but when he does something out of his usual character like cooking, it can be a sweet surprise for Bulma.
In the fanfiction, their relationship could be explored by showing how they adapt to each other. Vegeta's feral side might require Bulma to be more patient and understanding. And Bulma's needs might force Vegeta to become more domesticated in some ways. For instance, he might have to learn to control his temper around her family and friends. It could also show how their relationship evolves over time, starting from initial attraction despite their differences to a deep, loving bond that endures through various challenges.
At first, it might be a relationship based on mutual need. Vegeta needs Bulma's resources and intelligence to get back on his feet, and Bulma is intrigued by this alien. As time passes, they start to understand each other better. For example, Vegeta may start to open up about his past and his Saiyan pride, and Bulma will share her dreams and ambitions with him.