It could also be that he was chasing after a sense of adventure. Naruto was a very curious and energetic kid. He might have heard stories about the outside world and decided to run away from Konoha to explore and find his own path, away from the rules and restrictions of the village.
One reason could be the loneliness he felt in Konoha. People were often mean to him because he was the Jinchuriki of the Nine - Tails. He might have felt that running away was the only way to escape the constant ostracism and find a place where he could be accepted.
The village would likely be in chaos. Naruto, despite being an outcast in some ways, was still an important part of Konoha. The Hokage and the ninja leaders would probably send out search parties immediately to find him.
Another reason could be for the sake of protecting the village. Maybe he discovers some sort of threat that he believes he can only face by leaving Konoha. He doesn't want the village to be in danger because of him, so he runs away to draw the danger away from Konoha.
One possible reason could be the constant feeling of being an outcast in the village. Despite his efforts, many villagers still look at him with suspicion because of the Nine - Tails inside him. He might feel that running away would give him a chance to find a place where he is truly accepted.
One possible reason could be feeling ostracized. In the Naruto world, being different is often not easily accepted. As a female Naruto, she might face even more scrutiny and prejudice. Maybe she's tired of the constant expectations and the way she's treated in Konoha, so she decides to run away to find a place where she can be herself.
He could defect because of a false accusation. For example, if he is wrongly blamed for a serious crime in the village. This would make him feel betrayed and unwanted. Also, if someone he deeply trusts in Konoha turns against him, it might push him to leave. It's not easy for Naruto, who has always yearned for acceptance in Konoha, but such circumstances could drive him away.
He could be running away because of false accusations within the family. Just like how he was wrongly judged by the village in the original story, if his family wrongly blames him for something he didn't do, like stealing a family heirloom or causing harm to a family member accidentally, he might choose to run away rather than face their unjust wrath.
It could also be due to a lack of support. Naruto was constantly in trouble and trying to prove himself. If his family didn't show up in any way to support him and Konoha didn't do enough to encourage him in a positive way, but rather just punished him for his pranks, he might grow to hate them both.
One possible reason could be the constant scrutiny and pressure he faces in Konoha. Everyone has high expectations of him as the Jinchuriki of the Nine - Tails. He might feel suffocated by all the attention and the responsibility thrust upon him. Another reason could be the lack of true understanding from some of the villagers. Despite his efforts to protect Konoha, there are still those who look at him with distrust because of the Nine - Tails inside him.
Perhaps he feels pressured in Konoha. Having a kid adds more responsibility, and he might think that leaving Konoha will give him the space he needs to grow as a father and a ninja without the constant eyes of the village on him. He could be seeking a place where he can train his kid in a different way, away from the traditional Konoha methods.
It might be that his family is overprotective. Naruto is a very independent character. If his family in the fanfiction is constantly trying to hold him back from taking risks or going on adventures, he may feel suffocated. Running away would be his way of breaking free and exploring the world on his own terms.