No, it's not appropriate. Writing such fanfics involving sexual content between characters in a disrespectful or inappropriate way goes against moral and ethical standards, especially when it involves deities from mythology in a vulgar context.
One possible plotline could be that Annabeth discovers some sort of secret that Zeus is hiding, and a lemon (which could be a magical lemon or just a symbol) plays a role in uncovering the truth. For example, the lemon might be a key ingredient in a potion that reveals hidden memories.
Ares could also be a strong antagonist. He might view the alliance between Annabeth and Zeus as a threat to his own plans for power and chaos. He could try to start wars or create discord to break their alliance and stop whatever they are working towards.
One possible plot could be that Annabeth, in her quest for knowledge, accidentally stumbles upon a long - forgotten secret of Zeus. Zeus, initially angry, realizes her pure intentions and decides to guide her on a journey to uncover ancient mysteries that could change the balance of power in the pantheon. This journey would test Annabeth's wits and courage as she has to deal with Zeus' unpredictable nature and the challenges that come with the knowledge they are seeking.
You could write about their adventures together in a more family - friendly and respectful way. Maybe they team up to solve a problem in the realm of the gods, with no sexual undertones at all. Focus on their different powers and how they can combine to achieve a goal.
Well, you can try searching on fanfiction.net. There might be some works that involve Annabeth, Zeus and Lemon in interesting storylines. However, it might take some time to sift through to find the really good ones.
In such a fanfic, one possible consequence could be a major upheaval in the world of the story. The child would be of divine and demigod lineage, which might attract all kinds of unwanted attention from other gods and mythical beings. It could also lead to a power struggle as different factions might try to use the child for their own purposes.
It's inappropriate. The Percy Jackson series is a story that has a lot of young fans. Annabeth is a character that represents certain ideals and Zeus is a major deity in the mythology. Combining them in a 'lemon' (which usually implies sexual or overly suggestive content) context is completely against the spirit of the series. It would also be disrespectful to the source material and the many fans who enjoy the series for its adventure, friendship, and moral lessons.
The other gods would have mixed reactions. Some might be amused or see it as an opportunity to gain an advantage. For example, Hera, Zeus's wife, would be extremely angry and might seek revenge on Annabeth. The other demigods would be divided. Some might pity Annabeth, while others might see her as having overstepped or being too ambitious. The child, if it was born, would be a source of both hope and fear for different characters. It could be seen as a powerful ally or a dangerous threat depending on who you ask.
One common theme is the clash of different powers. Zeus is a powerful god and Annabeth is a smart demigod. When a lemon is added, it can be a symbol of something unique, like a test or a strange element that causes the clash of their powers even more. For example, the lemon might have powers that Annabeth wants to use but Zeus is against it.