Should I mention in my PhD application that I'm quitting my job in a month to spend time learning about the subject?
From the admissions perspective, it shows dedication but also chutzpah in a bad way. 

Many competitive programs have an admissions rate between 5-20%. Even if you are a strong candidate, it is not guaranteed that you will get in anywhere this year. This is why you have to apply to multiple programs as well as have a backup plan if you don't get in anywhere. Staying in a job (you might hate) would be such a backup plan and would show that you are someone who has Plan Bs.

If I read your application, I would have a nagging feeling at the back of my mind that you are rather sure of yourself and/or desperate and/or headstrong. None of these are good qualities for doctoral programs. 

There's also a slight feeling of being extorted - as in, if you don't admit me, I'm screwed. But that's not our problem, you put yourself in that situation, yet it still leaves an unsettling feeling.

Neither of these are enough to tank your application but it could sway someone on the cusp of denial/admission towards denial.

tl;dr: A much stronger application would be one that says, "if I am admitted, I plan on quitting my industry job in May and spending the summer ahead of matriculation preparing for graduate studies by doing xyz."