The main antagonists could be a cult that worships an ancient and evil deity. This cult might view Percy and Apollo's union as an abomination and try to stop it by any means necessary. They could be using dark magic and summoning all kinds of evil creatures to fight against Percy and Apollo. Also, there could be a power - hungry minor god who wants to break them apart so that he can gain more influence among the gods.
It could be the Titans who escaped from Tartarus and are seeking revenge. They might see Percy and Apollo's relationship as a threat to their plans of overthrowing the Olympians. Another possibility is some jealous gods or goddesses who don't like the idea of Percy and Apollo being mates.
Apollo himself could be seen as an antagonist in a way, because of his possessive behavior towards Percy. His actions that restrict Percy's freedom and independence are antagonist - like. However, in some cases, other characters who oppose Apollo's possessiveness might also be antagonists from Apollo's perspective. For example, if Percy has a close friend who tries to protect him from Apollo, Apollo might view that friend as an enemy.
One main antagonist could be Nyx, the goddess of the night. Since Apollo represents the sun, Nyx might see Percy as a threat as he is the champion of Apollo. She could try to plunge the world into eternal darkness and send her monstrous children to stop Percy.
Another possible antagonist could be some of the more power - hungry and jealous demigods. They might not like that Percy has been adopted by Apollo and is getting special attention. They could try to sabotage his missions or spread rumors about him to turn the other demigods against him.
Another theme is the idea of acceptance. Percy, coming from his own world, has to accept Apollo as a mate and vice versa. This is often shown through their internal monologues and interactions with other characters. For example, they might face opposition from some of the other gods or demigods who don't understand their relationship. And in dealing with this, they grow closer together.
Sometimes, other werewolves who don't accept Bella as Jacob's mate could be the antagonists. They might think that a human - werewolf relationship goes against their pack's rules or beliefs.
Often, other vampires or supernatural creatures could be the main antagonists. For example, vampires who are jealous of Damon's newfound mate might try to drive a wedge between them. Maybe there are some werewolves who see Bella as a threat to their territory because of her relationship with Damon.
Often, the Volturi can be antagonists. They have strict rules about vampire relationships and activities. If Bella and Jasper's relationship goes against their laws, the Volturi would oppose it. They might see it as a disruption to the vampire order they uphold.
One possible plotline could be Percy and Apollo facing challenges together in the world of the gods. For example, they might have to battle a powerful new monster that threatens both the mortal and divine realms. Another could be Percy adjusting to his new relationship with Apollo, dealing with the different nature of Apollo compared to his previous friends. Maybe there are also plotlines where they explore different mythical places together, like the lost city of Atlantis or a hidden grove in Olympus.
Often, the Volturi can be antagonists. They might not approve of Jasper and Bella's relationship for some reason, like they think it goes against vampire laws or traditions. Another possible antagonist could be a rogue vampire who has a vendetta against the Cullens and sees Jasper and Bella's relationship as a way to get to them.
One great Percy is Apollo's mate fanfiction could be 'Sunlight and Sea' where the author beautifully weaves their relationship. It shows Percy's connection with Apollo in a unique way, with adventures that bring out their chemistry.