Perhaps Nagini has her own motives. In fanfiction, she could be trying to atone for her past actions as Voldemort's pet. By protecting Harry, she hopes to find redemption. Also, she might see Harry as the key to a new future for the wizarding world, one where snakes and wizards can coexist more peacefully, and so she decides to safeguard him.
One reason could be a change of heart. Maybe Nagini realizes that Voldemort's cause is wrong and sees something in Harry that makes her want to protect him, like his kindness or his will to fight for good.
Well, in certain fanfiction scenarios, Nagini could be magically bound to protect Harry. Maybe a spell was cast that made her loyalty shift from Voldemort to Harry. So she would use her venomous attacks or her stealthy movements to keep Harry safe from Death Eaters or other dark forces. She might also be able to communicate with other snakes in the area and get them to help in protecting Harry.
Perhaps Voldemort discovers that there is a greater threat out there. In the fanfic, it could be that another dark force is emerging, and Voldemort sees Harry as an ally of sorts against this new threat. So, he decides to protect Harry in order to deal with this common enemy.
Another possibility is that he is in love with her. In fanfiction, love can be a powerful motivator. He might be attracted to her intelligence, her strong will, and her beauty. This love could override his loyalty to the Dark Lord and make him want to protect her at all costs. He could see her as a symbol of hope and a better future for the wizarding world.
Maybe they would talk about Voldemort. Since Nagini was Voldemort's pet and weapon, Harry might ask her about Voldemort's plans or weaknesses.
Maybe Gai has some sort of misunderstanding about Tenten and is acting aggressively towards her, so Naruto steps in to protect.
It could start with hostility. Harry sees Nagini as a tool of Voldemort, and Nagini is loyal to Voldemort at first. But then, some unexpected events might force them to interact more. For example, they could be trapped together in a magical place, and they have to rely on each other to get out.
One common reason could be his love for Lily. Since Harry is Lily's son, Snape might feel an obligation to protect him out of his undying love for her. It's like he's trying to preserve a part of Lily that still exists in the world through Harry.
In many Harry Potter and Nagini fanfictions, the relationship between them is often explored in unique ways. Some might depict Nagini as having a more complex personality than in the original books. For example, she could be shown as a victim of dark magic rather than just a tool of Voldemort.
Yes, there are some out there. You can try searching on fanfiction - specific websites like Archive of Our Own. Just be aware that the concept is quite unusual as Nagini is a snake and in the original series, not really a typical love interest for Harry.
One reason could be that the official Harry Potter canon is so extensive and detailed that it overshadows fanfiction. The original story has such a rich world, characters, and plotlines that it becomes the main focus for many fans, leaving less room for fan - created stories.