It could also be due to a past event that Naruto unknowingly was involved in. Maybe something bad happened to Sarutobi's family or friends and he blames Naruto for it, whether it was Naruto's fault or not. For instance, if there was an attack on the village and Sarutobi lost someone close, and he wrongly believes Naruto had a part to play in it, he would start to hate Naruto.
Another reason might be related to a past event that Naruto unknowingly caused harm to one of the Team 7 members' families. Even if it was an accident, the emotional pain could make the affected member, and maybe the whole team, hold a grudge against Naruto. For instance, if Naruto's actions during a battle led to the destruction of a family's home, and a Team 7 member was from that family.
It could be that Naruto has a different view of leadership than Sarutobi. Naruto might see Sarutobi's leadership as being too passive or too focused on maintaining the status quo. He, with his more energetic and revolutionary ideas, may resent Sarutobi for not being more progressive in making the village a better place for everyone, including himself.
Naruto might initially be confused. He has always looked up to Sarutobi as a respected elder, so being hated by him would be unexpected. He would probably try to find out why Sarutobi hates him. For example, he might go to Sarutobi and ask directly, hoping for an explanation.
One possible reason could be that he feels neglected. Maybe his family was always too busy with other things and didn't give him enough attention. This lack of attention during his formative years could have led to feelings of resentment.
One possible reason could be a misunderstanding. Maybe some false information was spread within the fanfic about the Uzumaki clan that made Naruto believe they were bad. For example, if there were lies about his family's past actions that he wrongly thought were true.
It could be that the fanfic portrays Hinata as being arrogant towards Naruto's past. Naruto has had a difficult childhood and if Hinata in the fanfic mocks or belittles that, it would make Naruto hate her. For instance, if she scoffs at his time as an outcast in the village and uses it as a means to elevate herself, Naruto would surely despise her for it.
One possible reason could be the way he was treated as a child. He was ostracized and shunned by many villagers because of the Nine - Tails inside him. This constant rejection and coldness from the people around him might have led to his hatred in the fanfiction.
Well, it could be related to Minato's absences. He was often away on important ninja business. Naruto might have felt abandoned by his father, and Kushina, seeing her son suffer in Minato's absence, could start to resent Minato. Also, there might have been some political decisions Minato made that didn't sit well with Kushina. For example, if he made an alliance that she thought was against the values of the village or their family.
It could also be due to a misunderstanding. Kakashi might be keeping some secrets from Naruto, like information about his parents or some hidden power within him. Naruto, being curious and eager for answers, could misinterpret Kakashi's motives for secrecy as a sign of distrust or dislike, which then turns into hatred.
Maybe in this fanfic, there has been a misinformation spread about Naruto that makes Kushina believe he has done something unforgivable. For example, if false rumors say that Naruto is involved in something evil, Kushina might start to hate him without knowing the truth.