One possible scenario is when Artemis criticizes Percy's fighting style. Percy has his own way of battling monsters, which is effective but might not be as refined as the way the hunters of Artemis do it. If Artemis were to publicly criticize his style in front of others, it would surely wound Percy's feelings. He might feel like he is not good enough in her eyes.
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.
He could become withdrawn. Percy might start to distance himself from Annabeth, not wanting to be around her if she's going to be cruel. He might go off on his own and think about what she said, trying to figure out if he did something wrong.
One possible reason could be miscommunication. Maybe Annabeth said something without realizing how it would come across to Percy. For example, if she was stressed about a quest and snapped at him when he was just trying to be helpful.
Well, in some fanfictions, Artemis might hurt Percy's feelings by rejecting his help. Percy, being the heroic type, offers his assistance in a dangerous situation involving the hunt or something related to Artemis' domain. But Artemis, in her pride and self - reliance, turns him down harshly, making Percy feel unappreciated and useless.
In some fanfictions, Artemis may hurt Percy because of her strict sense of justice. Say Percy breaks one of her rules unknowingly, like entering a sacred forest she has marked off. Artemis, being the goddess of the hunt and protector of the wild, might feel the need to punish him. This could lead to Percy getting physically injured, like being struck by one of her arrows. Or maybe Artemis is under a curse that makes her act out of character and she ends up hurting Percy without really meaning to. This then sets off a chain of events in the fanfiction where Percy has to deal with the injury and also try to find a way to break the curse on Artemis.
For Artemis, show her remorse. After she hurts Apollo, she could be filled with guilt. She might isolate herself from the other gods as she tries to come to terms with what she has done. She could also be shown as determined to make amends, perhaps by seeking out rare herbs or magical items that can heal Apollo. For Apollo, show his pain, both physical and emotional. He might be confused as to why Artemis would hurt him, but also be quick to forgive. He could use his musical and healing powers to try to soothe himself while waiting for Artemis to make things right.
It would be a complex dynamic. Artemis is known for her aloofness and strictness. Percy, on the other hand, is a bit of a rebel. So, initially, there would be friction. However, as Percy starts to learn from Artemis, like her hunting techniques and her connection to the moon, he would start to conform more. Artemis would in turn start to see Percy as more than just a wayward demigod and would protect him fiercely. They would form a bond that is both teacher - student and mother - son like.
Percy may become more reserved. With Artemis being possessive, he might start to hold back his normal, outgoing nature. For example, he may not be as quick to jump into new adventures or make new friends.