How to take a mental health related leave of absence (immediately before quals) without revealing mental illness to advisor
Have you considered asking for leave due to "urgent health-related issues" without providing more detail than you feel comfortable providing? There is no need to explain exactly what issues you are dealing with at the moment. It is perfectly sufficient to state convincingly -- perhaps with supporting documents from a medical doctor or other health professional -- that the issue is urgent and requires that you take leave as soon as possible. How you phrase this is something to discuss with your counselor.

Personally, I don't think it's reasonable, when your mental health is at stake, to agonize over what your adviser might or might not think of you. Chances are they will be understanding and appreciate that you come to them now and not later, when your health has deteriorated further and more work has piled up. By contrast, if your adviser turns out to be unsupportive or worse, it's time for you to look for someone else to work with! Also this is worth finding out sooner rather than later, so you may see this request as a litmus test of your adviser's qualities.

As another aside, brooding over other people's impression of you is a habit that isn't conducive to mental well-being. I suggest talking this over with your counselor as well, and finding a way to assert your needs and boundaries with confidence. It will come in handy later in your (academic) career.