Often in such fanfics, the theme of competition is present. Percy's jealousy could be a result of a perceived competition with another character for Annabeth's attention. This could be related to their heroic deeds. If another character is getting more praise or recognition from Annabeth, Percy might become jealous. It's a way to explore Percy's complex emotions towards Annabeth in a different light.
A typical scenario could be when Annabeth spends a lot of time with another male character while working on a project. Percy might see them getting along really well and start to feel jealous. For instance, if they are building something together in the fanfic, like a new weapon or a structure, and Percy feels left out.
Maybe some of Percy's mortal friends who don't really understand his relationship with Annabeth. They see how close they are and might be a bit envious of the special bond they share. For example, if Percy's friend was interested in Annabeth but knew he didn't stand a chance because of her and Percy's relationship, he could be jealous.
In many fanfictions, Annabeth might be jealous of Percy because of his popularity or his close relationships with other characters. It could also be due to his natural talent in certain areas that overshadows her own at times.
Communication is key. They need to sit down and talk about their feelings. For example, Percy could tell Annabeth that he's only being friendly with the new demigod who seems to like him, but his heart is with Annabeth. Annabeth could then share her insecurities. Another way is through a shared adventure. If they go on a quest together where they have to rely on each other, they'll realize that their relationship is stronger than any jealousy - inducing situation. They could also be given a task by Chiron that forces them to work together and face their jealousy head - on.
Annabeth is known for her intelligence and strategic thinking. She is a strong and independent character. Percy, on the other hand, is brave and has a strong connection to the water due to his father being Poseidon. In the fanfic, when Annabeth sits in Percy's lap, it might be a way to show their unique relationship which has developed over the course of their many shared experiences in the Percy Jackson universe.
The jealousy can initially strain their relationship. Percy's jealousy might make Annabeth feel smothered or like he doesn't trust her. For example, if Percy is jealous of Annabeth's friendship with a male demigod, he might act coldly towards her, causing her to be confused and a bit hurt.
One idea could be that a new demigod arrives at Camp Half - Blood. This new demigod is really good - looking and shows a lot of interest in Percy. Annabeth gets jealous seeing them together all the time, like during training sessions or at the campfire. Another plot could be that Annabeth has to work closely with an old acquaintance on a quest, and Percy gets jealous because he feels left out and he's not sure about this acquaintance's intentions towards Annabeth. A third plot could be that in a school setting outside the camp, Percy is getting a lot of attention from mortal girls because of his heroic looks, and Annabeth has to deal with her jealousy while also trying to keep their demigod world a secret.
Annabeth might be annoyed. She's very perceptive, so she quickly realizes Jason's jealousy. She'll probably tell him directly to stop acting childish and respect her relationship with Percy.
I'm not sure if there is a specific 'Percy and Annabeth bikini fanfic' available. There are tons of fanfics out there, but it would take some searching through fanfic platforms like Archive of Our Own or FanFiction.net to find one with that exact theme.