It could be Percy himself if he's acting unjustly. Even though he's beating up Hercules, if his reasons are not noble, like if he's just being full of hubris, then he would be the antagonist.
Well, it could be that Hercules has done something to Percy's friends or family. For example, Hercules might have accidentally injured one of Percy's close friends during a heroic adventure. Percy, being very protective, then decides to take matters into his own hands and beats up Hercules. Also, there could be a power struggle between the two, with Percy feeling that Hercules is becoming too powerful and a threat to his own world or community.
In this fanfiction, perhaps there is a misunderstanding between Luke and Percy. Luke might think Percy has betrayed him in some way, like siding with the wrong people in the context of their fictional world, and thus resorts to violence by beating him up.
Percy could be the instigator if he has a moment of rage or is misinformed. His impulsive nature might lead him to start the fight without fully understanding the situation.
Perhaps in the fanfiction, Percy is under some sort of dark influence. This could be from a new, unknown magical force. This influence makes him see Jason as a threat to his friends or himself, so he attacks Jason. Jason, on the other hand, is initially confused but then tries to fight back to defend himself while also trying to figure out what's wrong with Percy.
I'm not sure specifically about this particular fanfiction. Fanfiction can cover a vast range of scenarios, and while this is a rather unorthodox and not very positive one in the context of the characters, it might exist in some corner of the fanfiction world. You could try searching on popular fanfiction websites.
One similarity could be their heroic natures. In fanfiction, both are often depicted as trying to do the right thing. Percy is always saving his friends and the world from various threats, and Hercules is known for his heroic deeds. Another similarity might be their connection to the gods. Percy is the son of Poseidon, and Hercules is a son of Zeus in the original mythology, so in fanfiction this divine connection could be explored in similar ways, like having to deal with the expectations and powers that come from being related to the gods.
Most Percy fans would probably be outraged. Percy and Annabeth have a very special relationship in the original story, and fans are quite attached to it. This kind of fanfiction would seem like a betrayal of that relationship.
Yes, there are quite a few. You can find some on popular fanfiction websites like FanFiction.net or Archive of Our Own. Just search for 'Percy Jackson Beats Jason' and you'll get a list of relevant stories. Some might focus on a battle between them where Percy's skills and wits help him overcome Jason.