In a fanfiction where Percy Jackson goes to Forks, he might be really confused by the gloomy weather. He's used to more sunny and open places. He could meet the Cullens and there would be a clash of his demigod powers and their vampire abilities. Maybe he'd mistake them for some new kind of monster at first.
Perhaps a new, powerful demigod - vampire hybrid that Percy has to face. This hybrid could be causing trouble in Forks, and Percy has to stop it before it destroys the delicate balance between the vampires, werewolves and humans there.
Percy going to Ilvermorny would be a wild ride. First, he'd be amazed by the castle itself. The four houses would be a new concept for him. He might be sorted into a house based on his heroic nature, like Thunderbird perhaps. He'd introduce the idea of Greek gods to the students there, which would cause quite a stir. And his relationship with Annabeth would be put to the test as they are separated by different magical institutions.
One idea could be Percy getting involved in the vampire - human drama in Forks. Maybe he discovers the secret of the Cullens and tries to find a way to co - exist with them, given his own experiences with different species in his own world.
He'd probably have a hard time at first. The weather in Forks is very different from what he's used to. Also, the small - town atmosphere might be a bit suffocating for someone who's used to the more adventurous life of a demigod. But he'd adapt by using his skills. His athleticism from all his training as a demigod would make him stand out in school sports, which could be a way for him to make friends.
Well, one possibility is that Percy's demigod powers would create a stir in Forks. He might accidentally use his water - controlling abilities near the ocean or a river there and freak out the locals. Maybe he'd get involved in some sort of mystery related to the strange happenings in Forks that the Twilight characters are used to. He could also form an unlikely friendship with some of the Twilight gang, like Bella or Edward, despite their very different worlds.
A plot could involve Percy being sent to Forks on a quest. He discovers that there's a powerful artifact hidden in the area that could cause chaos if it falls into the wrong hands. The vampires and werewolves in Forks are also after this artifact for their own reasons. Percy has to navigate the complex relationships between these two groups while also trying to keep his true identity as a demigod a secret. Along the way, he forms an unlikely friendship with Jacob, who helps him understand the ways of the werewolves.
Another option could be a group of rogue demigods who have turned to the dark side. They follow Percy to Forks, hoping to find some powerful artifacts there. They don't care about the vampires or werewolves and are willing to cause destruction to get what they want. Percy has to stop them while also dealing with the local supernatural creatures.
Percy Jackson would probably stand out in Forks. He has this whole demigod aura and his powers are quite flashy compared to the more low - key vampires and werewolves. But he might adapt by using his wits. He could blend in by pretending to be a normal human, but his constant awareness of possible threats (which in Forks are vampires and werewolves instead of Greek monsters) would make him seem a bit paranoid to the locals.
He might meet some talented witches and wizards. Maybe there would be a student who is an expert in magical beasts and they could share stories about their encounters with strange creatures.