Maybe Artemis is on a hunt when she stumbles upon Percy. She might be initially cautious, as she's a goddess and Percy is a demigod. She could question him about his presence in Texas.
Well, they could go on a quest to find a lost artifact. Maybe it's a magical item that has the power to control the weather in Texas. They'd have to face all kinds of challenges like wild storms created by a rogue weather spirit and deal with some local mythical creatures that are guarding the artifact.
She would probably be furious. Artemis has a strong sense of justice and protecting the vulnerable. So, she would likely immediately start to plan how to deal with those responsible for Percy's abuse.
In some fanfictions, Artemis might react with intense anger. She is a protector, especially of the innocent. So she could immediately seek justice for Percy, using her power and influence among the gods to find the perpetrator.
The first encounter might be one of confusion. Percy could be startled to see Artemis suddenly appear in the woods. Artemis, on the other hand, may be trying to figure out why Percy is there. She might question him immediately, and Percy would have to explain his presence, perhaps stuttering a bit as he is in the presence of a major goddess.
Well, she could betray Percy by revealing his secrets to his enemies. Maybe she's forced to do so under some kind of magical compulsion or threat.
Artemis would probably be furious. As a goddess associated with protection, especially of women and girls, she would see this as a great injustice. She might immediately start planning revenge on the one who did this to Percy.
Percy could also react with a sense of determination. Instead of being defeated by the injury, he sees it as a challenge. He starts to figure out what made Artemis hurt him in the first place. If it was a misunderstanding, he tries to clear it up. If it was something more malicious, he starts to plot his revenge. He uses his own wits and the powers he has to either heal himself or find a way to make Artemis see the error of her ways. He might also seek the help of his friends, like Annabeth or Grover, to come up with a plan to deal with the situation.
Percy could interact with them in a variety of ways. For example, he might approach Athena in her temple seeking knowledge about the lemon. Athena, being the goddess of wisdom, could provide him with some cryptic clues. With Artemis, Percy could be on a joint mission where the lemon is somehow involved. Artemis, with her fierce independence, might initially be hesitant to work with Percy but then realizes they need each other.
It could be some rogue Titans who escaped from Tartarus. They want to cause chaos in Oklahoma and use Percy and Artemis to their advantage. Maybe they plan to capture them and use their powers to break free the other Titans.
They might go on a quest to find a lost artifact. Maybe it's a magical item that has the power to control the weather in Oklahoma. They could face various mythical creatures along the way, like a giant snake that guards the artifact.