In this fanfiction, it could be that Annabeth is influenced by her friends. They keep telling her how great Percy is and that she should make a move. So she plans a day out for them, like going to a carnival. As they are having fun on the rides, she looks at him and says, 'Percy, I've been thinking. I want us to be more than just friends. How about we go on a date?' This would be a really cute and fun scenario for the story.
Perhaps in the fanfiction, it's after a big battle. Annabeth realizes life is short and she doesn't want to waste any more time. So she arranges a quiet dinner for the two of them and asks Percy out during the meal. She would say something like, 'Percy, you've always been there for me. I want to explore more with you, just the two of us. Will you go out with me?'