And she is also written in first-person point of view. Considering that a character written in 1st person typically can share his/her most secret thought with the reader, would that be ok if my main character would also mistrust the reader until some point in the story ? I don't know exactly how I would implement that but... Basically, the result would be that she would keep her secrets for a while.
Unless you're going to constantly break the fourth wall, the character has no clue that the reader exists.
If you're writing in first person then the narrator is supposed to be the character. And so unless the character is one to self-censor her thoughts, the reader should see every thought the character has. I've heard from several booktubers how annoying it is when the reader is prevented from hearing the thoughts of a first-person (or deep third-person) protagonist. Maybe it could work if you were writing it in such a way that it felt like the character was talking to the narrator about what she was going through. But if the character is so mistrusting it might not make sense that she would sit down for such an interview if she didn't already trust the narrator.
I think this sounds like something you would have to be very skilled to pull off, if it is even possible. The narrator would probably need to be an unreliable narrator, and not just mistrustful but outright paranoid. Maybe from an angle of 'character is re-counting the story and has a paranoia spiral about who might be reading it' type deal. But what I'm vaguely picturing might not be what you are, so maybe my angle is all wrong too. With that said, I've found the best thing to do with stories is to write them. Once it's out of your head and on paper, you'll get a better sense if it works or not. I'd also advise not to get attached to this 'gimmick' (for lack of a better word). Go write your story, toss it in there if you want! But when you re-read for edits, be honest with yourself if it improves it or not.
The only way I can think of to pull it off would be to have her have multiple personalities that distrust each other. EX: Jill is the 'main voice', that distrusts everyone, 'Jane' is the 'reader', another one of the personalities, Jill does not trust Jane so hides, twists information. But even then I think this is going to be really hard to do well and not be really convoluted/confusing.
The Crow Road by Iain Banks is written in first person and has a lot of things that the MC only hints at for quite some time, dark secrets and all. Not sure if that counts, but every first-person book is a narrative even if you don't say outright "I don't think I trust you enough to tell you that bit yet." But I could swear I've read that very line somewhere, with no indication of how the narrator was imparting the information.
I think before asking this, it's best to ask who the protagonist is speaking "to," if that makes sense? If they're mistrustful of the reader, then it implies they're speaking to someone. Depending on how you go about, you don't necessarily need to reference or acknowledge who the "audience" is/who the protagonist is speaking to. Having an "audience" in mind might make it easier to navigate that. You could also have them be speaking to an inanimate object and slowly grow more comfortable speaking aloud to it. Steven Brust does something like this in his Vlad Taltos series. I forget which ones (one might be Tecla), but the main character is speaking to some kind of recording device in a few of the books, and he speaks to it like it's a real person at some points. Steven Brust does amazing work regarding first-person narration, so I always recommend reading his stuff. The Vlad Taltos series is sci-fi/fantasy though, just FYI.
As someone said before, that would make the narrator unreliable. And depending on what growth she has in the story. If she learns to trust people by the end of the story, then it would make no sense she wouldn't trust the reader. To tell you the truth, I hear stories all the time. When you work in insurance, you hear a lot of stories. And people, when they tell a story, even one about their life, do naturally not reveal everything. Because they're telling a story and they want to keep your interest. So it's not necessary for her to mistrust the reader in order to keep a few secrets.