In the fanfic, it could be that there are signs that Beckett is hiding something. Perhaps he has unexplained absences. He might say he's at work but Alexis can't reach him during those times. Also, he could be overly protective of his phone or personal space, which makes Alexis suspect that he's cheating.
Alexis might confront Beckett directly. She could say something like 'I think you're cheating on me. What's going on?' This is a very straightforward approach.
Perhaps there have been some rumors or whispers in their fictional world that have reached Alexis' ears. Even if they are baseless, they can plant the seed of doubt in Alexis' mind, making her think that Kate is cheating.
It could be that Castle is overprotective. He loves Alexis so much that even the slightest sign of Beckett not being on the same page as Alexis in Castle's eyes makes him think Beckett hurt her. Maybe Beckett gave Alexis some career advice that Castle thought was too pushy.
Maybe Alexis has a different vision of the original story or characters. Fanfic often takes creative liberties that might not align with Alexis's view of Beckett.
Maybe he saw Beckett spending a lot of time with another man without knowing it was for a case. Castle is a bit of a jealous type, and he might overreact to normal interactions.
She could be defensive of Beckett. Since Alexis probably has a good relationship with Beckett, she may try to convince Castle that he's wrong and that Beckett didn't mean any harm. For example, she might tell Castle about all the positive things Beckett has done for her.
Maybe in the fanfic, Beckett got a new job opportunity in a far - away place. This could make Castle think she's leaving him.
Kate might get defensive right away. She could say things like 'How could you think that of me?' and be angry at Alexis for not trusting her. This is a common reaction when someone is wrongly accused in a fanfic or in real life.
In the fanfiction world, miscommunication often plays a big role. Perhaps Harry was involved in some secret mission or task that he couldn't tell Ginny about right away. But Ginny, not knowing the real situation, might misinterpret his secrecy as cheating. For instance, he might be sneaking around to protect Ginny from a potential threat, but she doesn't see it that way.
A common theme is friendship. Many fanfics explore how Beckett and Alexis become friends and the bond they share. Another theme is growth, as they both develop as characters within the fan - created stories.