One possible reason could be to seek new adventures. They might feel they've reached a point in Fairy Tail where they've done all they can and want to explore new magic and challenges elsewhere. For example, they could have heard of a far - off land with a different type of magic that they are eager to learn about.
Perhaps Natsu is under the influence of a powerful and dark magic that makes him lose control of himself and accidentally hurt Lucy. It could also be that Natsu is trying to protect Lucy from something but his actions end up causing her harm instead. Like if he tries to keep her away from a dangerous situation by pushing her too hard and she gets injured.
In a fanfiction, there could be a situation where Lucy is forced to side with a group that Natsu is against. This would make Natsu see her as an enemy. Or perhaps there is a prophecy that Lucy is involved in that Natsu misinterprets as something bad. He might think she is a threat to the guild or his friends. Also, if there is a love triangle situation where Lucy chooses someone else over Natsu in a way that really hurts his feelings, that could be a reason for him to start hating her.
Perhaps they leave because of a disagreement within the guild. Maybe there's a new policy or a decision made by the guild master that they don't agree with. For example, if the guild decides to get involved in a war that they think is unjust, they might choose to leave rather than participate against their beliefs.
One possible reason could be for a practical joke. Lucy might be in a playful mood and think it would be funny to tie Natsu up and see his reaction. Another reason could be for his own safety. Maybe Natsu was about to rush into a dangerous situation without thinking, and Lucy tied him up to stop him. It could also be part of a training exercise. If Natsu needs to learn how to break free from restraints in combat situations, Lucy could be helping him by tying him up and making him figure out how to escape.
One possible reason could be that she feels she has outgrown the guild. Maybe she wants to explore new magic on her own and believes the Fairy Tail environment has become too comfortable and is hindering her growth.
One possible reason could be a conflict with other guild members. Maybe there's a misunderstanding or a fight that makes Natsu feel he no longer belongs. Another reason might be that he has a personal mission that goes against the guild's rules or values in the fanfiction. He could also be framed for something he didn't do and, feeling betrayed, decides to drop out.
One possible reason could be a misunderstanding within the guild. Maybe someone spreads false rumors about Natsu, leading others to believe he's done something wrong. Another reason might be a power struggle. If there are those in Fairy Tail who envy Natsu's strength and popularity, they could plot against him to get him out of the way.
Perhaps Natsu feels betrayed by the guild first. There could be a situation where he believes the guild has turned its back on one of his closest friends or on a principle he holds dear. This sense of being let down by the guild might lead him to take extreme actions that seem like betrayal. It could also be related to a power struggle. If there's a new power within the guild that Natsu sees as a threat to the guild's original values, he might try to oppose it in a way that looks like betrayal from the outside.
One reason could be that the fanfictions don't match his view of the Fairy Tail world. He has his own understanding of the guild, his friends, and their adventures, and when fanfictions deviate from that, he doesn't like it.
The story could focus on their journey to find a new place to belong. They might meet new allies along the way who have different values and magic abilities, which would add new elements to the story. For instance, they could encounter a group of mages who use music as their form of magic.