In some fanfictions, it could be related to power dynamics. Percy is often seen as a powerful and popular character. If he uses his influence or power in a wrong way towards Nico, like excluding him from a group activity or mission that Nico really wanted to be part of, it can be a reason for Nico to cry.
The story could develop with Percy realizing his mistake and trying to make amends. He might start by apologizing sincerely to Nico. Then, he could do something really nice for Nico, like helping him with a personal project or standing up for him when others are being mean.
One reason could be jealousy. Maybe Nico is jealous of Percy's popularity at Camp Half - Blood. Another reason might be a difference in their goals. Percy wants to protect the camp and the mortal world in a certain way, and Nico has his own ideas that don't align with Percy's. So, this divergence of interests could lead to a fight.
In a fanfiction, it could be that Percy gets influenced by some sort of magic or external force that makes him act coldly towards Annabeth. He starts ignoring her and when she tries to confront him, he says some mean things. Annabeth, who is used to Percy being her support, would be heartbroken and cry.
Well, perhaps Percy has some personality traits that don't mesh with what Nico is attracted to. Maybe Percy is too outgoing and boisterous for Nico, who might prefer someone more reserved.
One reason could be his regret over his past actions. For example, when he distanced himself from his family to pursue his career at the Ministry. He might realize later how wrong he was and cry out of remorse.
Maybe Annabeth says something really hurtful about Percy's past, like making fun of his insecurities regarding his father. That could make Percy cry.
One possible reason could be jealousy among other characters. Maybe there are those who envy Percy's powers and leadership, Jason's status as a hero, or Nico's unique abilities. Another reason might be a misunderstanding. For example, they could be wrongly accused of a crime they didn't commit, which leads to others turning against them.
In this fanfiction, perhaps Percy adopts Nico after Nico has been ostracized by the other demigods. Percy realizes that Nico needs a safe space. So he offers his home. As time passes, they start to understand each other better. Nico might open up about his past traumas to Percy, and Percy would use his own experiences to help Nico heal and feel like he belongs.
Well, it could be that Percy yells at Nico because he feels Nico is being too reckless in a dangerous mission. Maybe Nico is about to do something that could endanger the whole group, and Percy, being the protective type, loses his temper.
In a 'Percy and Nico are betrayed' fanfiction, perhaps they are betrayed by the gods. The gods, fearing their growing power, might spread lies about them among the demigods. Percy and Nico then have to fight not only against the false accusations but also the demigods who turn on them. They would have to find a way to clear their names and prove their loyalty.