The fanfiction might develop it through relationships. Say, Jason becomes closer to a group of characters that Percy wants to be part of. Percy then feels left out and his jealousy towards Jason intensifies. There could also be public events where Jason is praised and Percy is not, which is a common way to build up this jealousy plot.
Often, it starts with small incidents. For example, Percy might get praised for a simple task that Annabeth thinks she could have done better. This builds up her jealousy gradually.
Well, in this fanfiction, Jason's jealousy could be shown through his actions and words. He may make snide remarks about Percy's power. Maybe he tries to undermine Percy's achievements. He could also be seen avoiding Percy because seeing him use his powers makes Jason feel inferior. Like when they are on a quest together, Jason might not cooperate as well as he should just because he's envious.
Jason might show his jealousy through subtle actions like avoiding Percy during group activities or making snide remarks about Percy's powers when others are around.
The jealousy can create conflict. Jason might start to act out, causing rifts between himself and Percy or Thalia. This can lead to different story arcs, like a rivalry developing. For instance, he could try to undermine Percy's actions or try to get closer to Thalia in a way that seems forced, which drives the plot forward as other characters react to his behavior.
At first, it might show the tension between them due to Jason's jealousy. Jason could be distant or snippy with Percy. But as the story progresses, they might start to communicate. Maybe Percy notices Jason's jealousy and tries to reach out, which begins to thaw the ice between them.
Jason's jealousy might initially make him act out in negative ways. He could be bitter towards Percy and Thalia, which would isolate him from them and others. But as the story progresses, he might realize that his jealousy is unfounded and start to change for the better, learning to be more confident in himself.
It can create conflict. For example, Jason's jealousy might lead him to challenge Percy to a duel or some sort of competition. This then drives the plot forward as the other characters have to deal with the fallout of their rivalry.
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.
Often, it starts with small incidents. For instance, Bella might get praised in front of Lauren for something she did. This makes Lauren feel inadequate. Then, as the story progresses, more and more situations arise where Bella is in the spotlight and Lauren is left out. Like at a big event in the fanfiction, Bella is the center of attention while Lauren is just a bystander.
One reason could be Jason's popularity. Maybe Jason is getting more attention from their peers or the gods, which makes Percy jealous. Another might be related to a girl they both like. If the girl shows more interest in Jason, Percy could feel jealous.