Yes, there might be such fanfics out there. The idea of combining the Naruto universe with the concept of a Grimm from another series is quite creative. Fanfic writers often like to cross over different fictional elements to create unique and interesting stories.
In a fanfic like this, there could be a lot of conflict. The Grimm's presence could be seen as a threat by the Hidden Villages. Naruto would then have to convince everyone that his pet Grimm is not a danger. Also, the Grimm might have special abilities that Naruto could use in battles or to solve problems in the village. For example, if the Grimm can fly, Naruto could use it to quickly reach far - off places or spy on enemies. And Naruto could also teach the Grimm some of his ninja values, like loyalty and perseverance.
Maybe there are other Grimm that see the pet Grimm as an outcast for being with a human. These Grimm could come after the pet and Jaune, making them the antagonists. They would try to either kill the pet and bring it back to the Grimm world or kill both Jaune and his pet to keep their world 'pure'.
Well, perhaps the plot could involve Jaune trying to train the pet Grimm. He might face challenges from other characters who are against having a Grimm as a pet due to their inherently dangerous nature.
I'm not sure exactly as I haven't read this specific 'naruto first grimm fanfic'. It could be a fan - made story that combines elements from Naruto and Grimm. Maybe it has Naruto characters in a Grimm - like world or vice versa.
Perhaps they go on an adventure to find a special kind of herb that can heal the sick in the village. The lion's keen sense of smell helps them track it down, and Naruto uses his ninja skills to fend off any threats that come their way during the search.
Well, one possibility could be that Naruto enters the world of Grimm. Since he has his ninja skills, he might initially be confused by the Grimm but then start using his jutsus to control them. Maybe he uses his shadow clone jutsu to surround and corral the Grimm, turning them into an army of his own. He could use his chakra to influence their behavior, like making them attack his enemies instead of the innocent.
You could try searching on popular fanfic websites like Fanfiction.net or Archive of Our Own. Just type in 'naruto first grimm fanfic' in their search bars.
One possible fanfic could be where Naruto and Sakura find an injured pet in the forest. They decide to take it back to the village and nurse it back to health. As they care for the pet together, their relationship also deepens, with Sakura seeing Naruto's kind heart in a new light and Naruto admiring Sakura's nurturing side.