Lore can be seen as the back story. It might include details about the origins of a place, the past of characters, or the hidden histories that shape the current situation in a story or game.
The relationship can start with mutual distrust. Percy has seen Hera's manipulative side before, and Hera might not fully trust Percy at first. But then, they could be forced to work together on a task. For example, saving a group of demigods from a powerful enemy. As they face challenges side by side, they begin to respect each other's abilities. Percy realizes that Hera has her reasons for her actions, and Hera sees Percy's courage and loyalty.
Perhaps in a fanfiction, Zeus, in a fit of anger over Hera's interference in one of his mortal affairs, uses his power to ostracize Hera from the other gods for a time. This makes Hera feel isolated and hurt, and she has to find a way to regain her standing while also dealing with the betrayal of her husband. It could show her strength as she overcomes this difficult situation.
Legend of Olympus novels were relatively rare because time travel novels were usually based on modern web novels, while Legend of Olympus was a type of ancient Greek mythology. However, some novels might use the legend of Olympus as the background to construct a world similar to ancient Greek mythology. For example, some novels might tell the story of a modern person traveling to ancient Greece to become a hero or god. This type of novel is relatively rare. If you are interested, you can try searching for it.
The Lore TV show is indeed fictionalized. It takes elements from various sources and weaves them into fictional narratives that may or may not have real-world inspirations, but are not based on actual events.
Percy would probably be very cautious around Hera at first. He might be a bit sarcastic too, like 'Great, the Queen of the Gods is here. What could go wrong?' But Hera would likely be all business, trying to get Percy to do her bidding in a very matter - of - fact way.