Yes, there are some out there. You can try searching on popular fanfiction websites like Archive of Our Own or Fanfiction.net. Just be aware that the quality can vary greatly. Some might focus more on the relationship development between Percy and Carlisle, while others could be part of a larger crossover plot with other characters from both universes.
The main themes usually revolve around adaptation. Percy has to adapt to the rules and lifestyle of the vampire world, which is very different from his own. Carlisle, on the other hand, has to understand Percy's connection to the Greek gods and the whole demigod culture. Another theme is the forging of a new family. Since they are mates, they create a unique bond that might not be fully understood by either of their original families or groups. And love in the face of differences is also a big theme, as they come from such disparate fictional backgrounds yet find a connection with each other.
It could start with an unexpected encounter. Perhaps Percy somehow ends up in the Twilight world or vice versa. Their relationship might develop slowly as they try to understand each other's worlds. Carlisle, being an old and wise vampire, could be intrigued by Percy's youth and his connection to the gods. Percy, on the other hand, would have to deal with the fact that Carlisle is a vampire, which is different from the monsters he's used to in his own world. And as their relationship progresses, they would face challenges not only from their own worlds but also from the differences in their natures.
You could try searching on popular fanfiction websites like Archive of Our Own (AO3). There's a large collection of fanfics there, and you can use the search function to look for this specific pairing. Just type in 'Percy Jackson Carlisle Cullen mate' and it should show relevant results if any exist.
One common theme is loyalty. Both Percy and the werewolf characters often show loyalty to their friends and mates. Another theme is the struggle between different worlds. The werewolves come from their own world, and Percy from his demigod world, and there's often a struggle to co - exist.
One common theme is loyalty. The werewolf mates often show extreme loyalty to each other and to their friends in the Percy Jackson world. Another theme is the struggle between different natures. The werewolves have their wild side, and the demigods have their own powers and responsibilities, and the fanfictions often explore how they deal with these differences.
One possibility in such a fanfiction could be that their relationship completely alters the dynamics of the original Twilight story. For example, it might change how the Cullen family functions and their interactions with other characters. Maybe Bella's connection with Carlisle would lead to a different approach in dealing with the Volturi.
Well, it could be about the challenges they face in raising the pups. Maybe there are enemies who target the pups because of their special powers or lineage related to Percy Jackson.
Another aspect could be the exploration of the vampire - human relationship in a different light. With Bella and Carlisle, it's not the typical young vampire - young human love story like Bella and Edward. It might delve deeper into the more mature and complex aspects of a relationship between a much - older vampire and a human.
Love and devotion are very common themes. These fanfictions often explore how Carlisle and his mate are deeply in love and devoted to each other, through various situations.