Well, he could experience intense pain and weakness. Depending on the type of poison, it might affect his connection to his father Poseidon. Maybe he'd be unable to communicate with sea creatures or draw strength from the ocean as he usually does. His friends would be in a panic. They'd probably split up, with some staying to take care of Percy while others go on a quest to find the source of the poison and a cure. Chiron at Camp Half - Blood would also be involved, providing what knowledge he has about such poisons.
Perhaps it's a gift from his father, Poseidon. Since Poseidon has control over the sea and all things related to water, he might have some connection to a power that gives Percy wings, like as a way to help Percy reach new heights both literally and metaphorically.
In the fanfic, it could be that Percy gets Triton as a gift from Poseidon. Triton could be initially reluctant to be with Percy, seeing him as an upstart half - blood. But as they spend time together, they start to understand each other better. Triton might help Percy improve his control over water, especially in more complex and powerful ways. They could go on quests together, like finding a lost ancient sea artifact that has the power to either save or destroy the world of the sea gods.
There shouldn't be any legitimate fanfiction with such a plot. The Percy Jackson series has a certain set of rules and moral and ethical boundaries within its fictional universe. The idea of Percy getting Hera pregnant is not only out of character but also violates the general tone and values of the series.
In such fanfictions, Percy's character is often explored through his resilience. For example, if he's physically abused, we see how he still tries to fight back or find a way to escape, which shows his determination. His loyalty also comes into play as he might not want to burden his friends with what he's going through.
He might become an even more powerful figure. The trident could enhance his abilities to control water and perhaps give him new powers related to the sea. He could use it to defend his friends and fight against stronger enemies.
Well, in a 'Percy Jackson Gets Arrested' fanfiction, perhaps he could be framed by a monster in human form. The police, not knowing his true identity as a demigod, might take him into custody. This could lead to a whole lot of chaos as his friends from Camp Half - Blood would surely try to break him out. They would have to figure out how to deal with the mortal legal system while also fighting off any monsters that might be taking advantage of the situation.
Perhaps he gets the power of astral projection. He could send his spirit out of his body to explore areas that are otherwise inaccessible or to spy on his enemies. This would be a very useful power especially when dealing with secretive foes or trying to find hidden places in the mythological world.
Writing about underage characters getting pregnant is inappropriate and goes against the spirit of the original 'Percy Jackson' series which is mainly aimed at a younger audience. So, there should not be any such fanfiction in an appropriate context.