Maybe there is an ancient evil that has been awakened by the presence of two immortal wizards. This entity could be a powerful force that wants to destroy all of magic, and Percy and Harry are the only ones who can stop it. Their immortality gives them the time and the ability to learn about this ancient evil and find its weaknesses.
There could also be a wizard from their own past who was jealous of their abilities even before they became immortal. Now that they are immortal, this wizard's jealousy has turned into full - blown hatred and he plots to make their immortal lives a living hell by constantly causing trouble for them and their friends.
It could be a group of dark wizards who see Percy and Harry's immortality as a threat to their own power - hungry plans. They might try to eliminate them to gain more control over the wizarding world.
It could be an old enemy from Greek mythology that has been resurrected. For example, a powerful Titan who has a vendetta against the demigods, especially Percy and Annabeth.
A powerful dark wizard who has made a pact with a Titan. This antagonist could be using dark magic to corrupt the boundaries between the wizarding world and the world of the Greek gods. He wants to gain ultimate power by controlling both realms, and Harry and Percy would have to stop him.
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.
Maybe it's the Titans who are still lurking in Tartarus and don't want Percy and Zoe to succeed in whatever mission they have. The Titans could see them as a threat to their potential future plans of rising again.
One common antagonist could be an old enemy of Artemis who doesn't approve of her marrying Percy. This enemy might be a powerful Titan who sees their union as a threat to his plans for revenge against the gods.
It could be the navy. They see pirates as outlaws and would be determined to capture Percy and his crew. They might have better - equipped ships and more trained soldiers, making them a formidable foe for Percy and his fellow pirates.
Monsters could also be antagonists. With Percy in a new, more powerful state as a godling, there could be new and more powerful monsters that are drawn to his power. These monsters might be stronger and more intelligent than the ones he faced as a demigod, and they could be after his godly essence or trying to use him for their own dark purposes.
In some fanfictions, it could be an ancient and powerful Titan who has escaped from Tartarus and is seeking revenge on the demigods. This Titan might see Percy and Thalia as the main obstacles in his path to domination.
A powerful sorcerer who wants to use Percy for his own dark purposes could be an antagonist. This sorcerer might see Percy's unique abilities as a key to unlocking some sort of forbidden magic.
Often, it could be old enemies like the Titans. In fanfiction, they might return in new and more devious ways. For example, Kronos could have a new plan to break Percy and Annabeth apart before trying to take over the world again.