There are likely some out there. You could try searching on fanfic platforms like Archive of Our Own. They have a large collection of fanfictions and you might be able to find one with this particular pairing.
One possible plot could be that Luke initially starts out as a rival to Percy but as they interact more, he begins to see Percy's true self and falls for him. Maybe they are forced to work together on a dangerous quest and during that time, Luke's feelings develop.
I really liked 'Luke and Percy: A New Beginning'. In this fanfic, the author imagines a scenario where Luke gets a second chance. Percy is initially skeptical but as the story progresses, they form a bond that is both unexpected and heartwarming. It's a great read for those who want to see a different side of their relationship.
A common theme could be redemption. Since Luke made some bad choices, a fanfic could focus on his journey towards redemption. Percy could play a key role in this, either by being the one to offer forgiveness or by being the example that makes Luke want to change.
One aspect that could be explored in a Percy Jackson and Luke fanfic is their opposing views. Percy stands for the good, while Luke turns to the dark side. A fanfic could delve into what might have made Luke change his path if Percy had intervened earlier. It could show their early interactions at Camp Half - Blood and how Percy's loyalty and Luke's growing resentment were seeded.
Percy might love fanfiction because it offers new and exciting adventures that are different from his own but still related to his world. It gives him a chance to see different interpretations of his friends, himself, and the events he's been through.
The writing style also matters. If the author can create a vivid world and bring the characters to life with their words, it makes for a great fanfic. Using descriptive language to show the emotions of Luke and Percy, like their anger, fear, or hope, can really draw the reader in.
One common theme could be the struggle between good and evil as Luke has his own complex relationship with the darker side in the original story and Percy is more on the side of good. Another might be friendship turned complicated due to different loyalties.
In a fanfiction about Percy falling in love with Aphrodite, it would be a story full of drama. Aphrodite is a complex character, and Percy's love for her could be both a curse and a blessing. He might be attracted to her beauty and allure at first. But then he would have to deal with the jealousy of other gods and the expectations that come with being in a relationship with a powerful goddess. Also, his friends might not understand his choice, which could create rifts in his relationships.
They could have been friends. Maybe they played together in the neighborhood or had some shared experiences like going to the same school. They might have bonded over things like a mutual dislike for a bully or a love for a particular sport.