One possible plot could be Percy getting caught in one of Artemis' hunts by accident. They initially clash due to their different natures but gradually develop feelings for each other.
Perhaps Percy is cursed in a way that only Artemis can break. They are forced to be in close proximity while trying to figure out the solution. Along the way, Percy discovers a side of Artemis that no one else has seen, and she sees his determination and selflessness. Their relationship starts as a reluctant alliance but slowly turns into love. There could be many obstacles like the other gods not approving, or Artemis' own doubts about getting involved in a romantic relationship, which would add more drama to the fanfiction.
In some fanfictions, Percy might be depicted as Artemis' secret lover. Fanfiction allows for a lot of creative interpretations of characters and their relationships. However, in the original mythology and main series, this is not a canonical relationship.
One possible plot could be that Artemis is captured by some powerful ancient enemy, like a primordial being. Percy, with his skills and courage, embarks on a quest to find her. He might face various challenges and monsters along the way. Maybe he has to solve some riddles related to the moon, as it's Artemis's domain. Finally, he locates Artemis and uses his water powers in a creative way to defeat the captor and free her.
One possibility is that Artemis has been misinformed. Some enemy of Percy has tricked her into thinking that Percy is a danger to the demigods. So she kidnaps him to stop him from causing harm. But as she keeps Percy captive, she slowly realizes that she has been deceived when Percy starts to explain his side of the story and the real threats he has been fighting against.
A plot could be that there's some sort of dark magic that is affecting the balance between the gods and the demigods. Artemis and Percy are brought together to combat it. Along the way, Percy shows a side of himself that Artemis is drawn to. His courage and selflessness intrigue her. And in a moment of vulnerability, Artemis kisses Percy. This kiss then becomes a catalyst for a deeper exploration of their relationship and the greater threat at hand.
Yes, there are some. You can try searching on fanfiction websites like Archive of Our Own. Some writers might have explored this unique relationship in their stories.
One possibility is that Percy grows up with a strong sense of independence and a connection to nature. Since Artemis is the goddess of the hunt, Percy might be trained in hunting and survival skills from a young age. He could also have a special bond with the animals in the forest, being able to communicate with them to some extent. This upbringing would likely make him a formidable fighter, not just with a sword but also with bows and arrows and stealthy combat techniques.
In this fanfiction, it could be that Artemis is in a situation where she is about to be killed or lose her powers forever. Percy, being the selfless hero he is, takes on a deadly task to prevent that from happening. Maybe it involves facing an ancient and malevolent god who has a vendetta against Artemis, and Percy uses his unique abilities and the element of surprise to try and stop this god, but at the cost of his own life.
One possible plot could be Percy struggling to fit into Artemis's group of hunters. Since he's male and the hunters are usually female, there would be a lot of conflict and misunderstanding at first. He might have to prove himself worthy of being part of the group, not just because he's Artemis's adopted son. He could go on a quest to retrieve a sacred item for Artemis to show his loyalty.
One possibility is that they are forced to share a bed due to some magical or dangerous situation. Maybe they are in a strange place where there is only one bed available as shelter from a threat. They start off awkwardly but gradually learn to respect each other more during the night.
One possible plot could be that the son of Artemis discovers a threat to the demigod world that is related to his mother's domain. He then seeks out Percy for help because of Percy's reputation. Together, they embark on a journey to find a magical artifact that can stop the threat.