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.
She could also try to be understanding and ask Alexis why she has such a suspicion. By having an open conversation, she hopes to clear up any misunderstandings. In the fanfic, this might lead to a deeper exploration of their relationship.
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.
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.
The fanfiction could be set in a school where the narrator is a student who idolizes Kate Alexis. They see her as a role model. One day, Kate Alexis visits the school for an event. The narrator is in awe and finally gets the chance to meet her. They simply say 'I love you' and then go in for a hug, which is a moment of pure joy for the narrator.
One possible fanfiction could be about a person who has a deep admiration for Kate Alexis. They might meet by chance in a park. The protagonist feels an instant connection and the urge to hug her, but is too shy at first. As they start talking, they discover shared interests, and the love for Kate Alexis grows stronger. Eventually, the protagonist musters the courage to give her a warm, friendly hug.
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.
She might confront Harry directly. In the heat of the moment, she could burst out with her suspicions and demand an explanation. For example, she might say, 'Harry, I've noticed you've been acting strange around that girl. Are you cheating on me?'
I think Gibbs would be a bit protective. He might give Dinozzo some fatherly advice and also make sure that Kate is taking it easy at work. Abby would be super excited and would likely start knitting baby clothes right away. McGee would probably be a bit shy about the whole thing but still offer his congratulations.
A lawyer could be an important character. This lawyer takes on Alexis's case, believing in their innocence. The lawyer has to battle a corrupt legal system or powerful opponents who want to see Alexis locked up for good. The fanfic could show the lawyer's strategies and how they build a case in Alexis's favor.
I'm not sure specifically as there could be many different interpretations. It might be a fan - created story where a character named Castle believes that Kate is cheating in the context of their relationship within the fan - created scenario.