The thing about Roronoa Zoro and Perona's relationship in One Piece is that it is unique and interesting, to say the least. It developed during the two-year timeskip within the series. What was at first enmity gradually transmuted into a somewhat unexpected alliance and even a form of friendship as time wore on with the two on Kuraigana Island.
Probably one of the most remarkable aspects of their relationship is just how much time was spent together. Fans have actually mentioned that Zoro has spent more time with Perona during the timeskip than he previously had spent with the Straw Hat crew. This extended period of interaction gave a very unique opportunity for character development and relationship building outside of the usual crew dynamics for Zoro.
Throughout his time spent at Kuraigana Island, it was a tale of two extremes with Zoro and Perona. One minute, they were hostile, and the next, they were bringing their heads together with a mutual trust and dependence to cast off different enemies. It was then really the forced acceptance of cohabitation that moved their relationship status from enemies to being friends, or as close as respect allows.
Where Zoro always played the role of rigid and stern, Perona was more playful than that, with a flavor of gothicism in her character. This caused several humorous interplays, notably when regarding Zoro's inability to find his way around. To both the manga and the anime, like Nami, Perona called Zoro a "fantasista" for being able to bear no sense of direction. These moments of levity helped to humanize Zoro's character and showcased a different side of him outside of his usual serious demeanor.
And that was also the time he spent with Perona, or at least some of the time. Now it was all part of Zoro's character development, but the training with Mihawk really didn't have anything to do with it. Perona was no part of the training architecture, but more or less, she represented a special kind of background on the island and around Zoro during this important character development phase.
Although the whole concept of Zoro and Perona's relationship interests fans so much, the history between the two somehow remains platonic at best in the series. Few fans have even theorized, or rather tried to draw conclusions, on potential romance between the two, but those remain at best a fan speculation sans official backing.
Later chapters of the manga once more got her attention from the fandom because of her involvement with Zoro. Perona and Zoro shared experiences during the timeskip, which made their relationship special compared to other series with other characters.
This is not to say that, despite the great time Zoro spent with Perona, his main loyalty holds to his crew of the Straw Hat pirates. Zoro had always remained as Luffy's right-hand man in crucial roles, and his place in the crew had always been a fighter, almost vice-captain, no matter how many character developments he underwent in those two years.
Another interesting case study would be the dynamic of Zoro and Perona. Their development from enemies to begrudging allies shows one of the greater complexities relationships can hold in the world of One Piece and how situations may bind together people who would otherwise have no contact with one another.