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.
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 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.
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.
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 Alec noticed Magnus being secretive about his phone or meetings. If Magnus is suddenly cancelling their plans often without proper explanation, it could make Alec suspicious. Also, if Magnus is overly friendly with someone else in a way that seems inappropriate, like getting too close or sharing personal things that he usually only shares with Alec.
Maybe she saw Harry spending a lot of time with another girl without proper explanation. In a fanfic, it could be that Harry is often with a new female character in his study or on some secret missions, and Ginny misinterprets this as something more than friendship.
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.
One possible reason could be that Magnus has been spending a lot of time away from Alec. Maybe he has been staying out late at night without a proper explanation. Another reason might be that Magnus has been secretive about his phone or messages. Alec could have seen Magnus being a bit too friendly with someone else, which made him start to suspect cheating.
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.