One possible motive could be a misunderstanding. Maybe Hiashi mistakes Naruto for an enemy who has infiltrated the village in disguise. Another reason could be related to clan politics. If Naruto somehow gets involved in a situation that threatens Hiashi's clan interests, Hiashi might take extreme measures.
Hiashi could be initially skeptical. Naruto has a reputation for being a bit of a troublemaker at times. But if Naruto presents a good case, Hiashi might warm up to the idea. For instance, if Naruto shows how his actions would bring honor and prosperity to the Hyuga clan.
Naruto might convince Hiashi by pledging his loyalty to the Hyuga clan. He could say that he'll uphold the clan's values and traditions. Also, Naruto could mention how Hinata has been a positive influence on him, which shows his respect for her. He might also talk about his plans for the future and how Hinata will be an important part of it in a way that benefits both the Uzumaki and Hyuga clans.
In this fanfic, Naruto might save Hiashi in a battle situation. Maybe Hiashi is facing a powerful enemy that he can't handle alone, and Naruto steps in with his strong ninja skills. For example, if Hiashi is being attacked by a group of rogue ninjas, Naruto could use his Rasengan to fend them off.
One possible reason could be that Naruto is in love with a Hyuga clan member. Since Hiashi is the head of the Hyuga clan, Naruto might think it's proper to ask for his blessing. For example, if he has feelings for Hinata, Hiashi's daughter. Hiashi's approval would mean a lot in terms of family and clan acceptance.
In Naruto and Hiashi fanfiction, Naruto and Hiashi are the central figures. Naruto is often depicted as the catalyst for change or a source of new ideas in these stories. Hiashi represents the traditional values and power of the Hyuga clan. Sometimes, other characters from the Naruto universe might also be involved, like Hinata, who is Hiashi's daughter and has her own connection to Naruto. But overall, Naruto and Hiashi are the main focus, with their interactions driving the plot forward, whether it's about mending clan relations, fighting a common enemy, or exploring new ninja techniques.
Maybe in the fanfic, there is a dark force that tries to manipulate Hiashi into attacking Naruto. The fanfic could explore how Naruto tries to reach out to Hiashi, to show him that they are being played. Along the way, they have an intense battle where Naruto shows off his new jutsus and Hiashi uses his clan's most secret techniques. In the end, they realize they were on the same side all along.
One interesting Naruto and Hiashi fanfiction could be about Hiashi training Naruto in some secret Hyuga techniques. Maybe Naruto has to prove himself to Hiashi first, and through a series of trials, they form an unlikely bond.
Well, it could be about Naruto growing up in the Hyuga clan. Maybe he discovers his true parentage early on and has to deal with the expectations and politics of the clan. He might learn the Gentle Fist technique in a different way than Neji or Hinata.
One possible plot could be that Naruto gains a whole new family structure. Hiashi might train Naruto in the ways of the Hyuga clan, which could make Naruto an even more formidable ninja. He would also be exposed to the Hyuga's strict traditions and might have a different view on the ninja world. Maybe he'd form a closer bond with Hinata earlier as they'd be siblings in this scenario.