Yes, there are some out there. You can try looking on fanfic websites like Archive of Our Own or FanFiction.net. Just search for 'beerus x bulma' and you'll likely find a variety of stories with different takes on their relationship.
The contrast between their characters. Beerus is a powerful god of destruction, while Bulma is a brilliant human scientist. Their different natures can create really interesting dynamics in a fanfic.
In the 'beerus vs omega shenron fanfic', Beerus would likely start the fight with his confident and somewhat arrogant attitude. He might underestimate Omega Shenron at first. However, Omega Shenron's ability to absorb energy could give him an edge. If Beerus doesn't quickly realize the threat and use his full power, he could be in trouble. But ultimately, Beerus' experience as a God of Destruction might give him the upper hand in this fictional battle.
Their relationship could be developed through shared interests. For example, both are interested in unique and delicious food. Bulma could cook special dishes for Beerus, which would gradually make Beerus see her as more than just an ordinary Earthling. He might start to respect and even protect her in return.
Maybe there's a plot where Beerus has a premonition about a threat to the universe that only Bulma's technology can help prevent. So, he has to team up with her. Along the way, they learn more about each other's worlds. Bulma gets to understand the power and responsibilities of a god like Beerus, and Beerus discovers the amazing potential of Earthling technology. This could lead to a deeper friendship or mutual respect between them.
One possible plotline could be that Broly, in his rage, accidentally provokes Beerus. Broly, not knowing who Beerus is, attacks him. Beerus, initially amused, decides to teach Broly a lesson but soon realizes that Broly's power is growing at an alarming rate. This leads to an intense battle where Broly pushes Beerus to use his full power. Another plotline could be that Beerus hears about Broly's power and decides to test him. He travels to Broly's location and challenges him. Broly, seeing Beerus as a threat to his friends or his planet, fights back with all his might.
In a 'Goku vs Beerus' fanfiction, it's really hard to say who would win. It could depend on how the author wants to write it. If the author wants to show Goku's growth, Goku might win through some new transformation or power - up. But Beerus is a powerful God of Destruction, so he also has a great chance of winning.
One idea could be that Beerus is intrigued by Bulma's intelligence and her ability to create amazing inventions. He might start visiting her more often, causing some comical misunderstandings among the Z - fighters. For example, Vegeta could get jealous of their interactions.