There are two questions here; the question you asked and the question you meant to ask.
CAN you apply for unemployment? Sure you can. There are no circumstances I can think of under which someone is going to refuse to allow you to apply.
But I think what you meant to ask is, Will you be approved for benefits? And that's a different story. While you have not told us much about what this person was doing, it is VERY rare that you can quit your job and still collect. Unless this person was indulging in some kind of illegal harassment towards you, the chances are that you will not be able to collect.
So while you can apply, in the sense that it is not illegal to do so and the worst that can happen is that the state says no, on the basis of what is in your post it's unlikely that you'll get it.