User blog comment:SayuriDarling/The Mark of Athena Official Discussion/@comment-4649443-20121019040618/@comment-3146930-20121105215143

From the way Annabeth starts to describe it, Athena falls in love with a man and they have a connection of the mind instead of the body. So instead of Athena actually sleeping with a guy, they connect with their thoughts. From this, a child is formed inside Athena's mind, possibly like how normal children are grown in the uterus. Anyway, once the child has grown inside Athena's head, Athena most likely splits her head open or something similar and delivers the child to the father as a gift.

As far as the series goes, it has never been stated if Annabeth is older or younger, as even a hint of her birthday has never been mentioned by Rick in any way (books, blogs, website, ect). As far as the link the person above posted, I've never really trusted that site as many of the quotes or information about the characters are very strangely written or outdated. It might be true, but I just don't trust that source.