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."

I wouldn't mention quitting your job, because there's no need to do so. Instead focus on the extra things you're doing to prepare yourself without mentioning your current work.

I did something very similar when I applied for PhD programs in economics. I took two years off after undergrad before starting my PhD, and while I did work, one of the main things I did was read seminal literature in my field - Adam Smith, Keynes, etc. I talked about this in my application, and knowing my department as I do now, I'm entirely certain this played a large role in getting me in.

This is especially true if you've done your homework on what things the department you're applying to is likely to care about. In my case economic history was a very important field for them, so it mattered.

So, don't mention quitting your job but do mention extra things you're doing, even if it's as simple as reading. I actually wouldn't say something like "if admitted I intend to..." as @RoboKaren suggested; doing this extra stuff because you're passionate about it and not because you know for certain you'll need it for school carries extra weight. "If admitted I plan to read the Wealth of Nations!" is very different from "While working in industry after undergrad I spent my free time reading the Wealth of Nations."

I look at graduate applications each year, though not in your field. I think that the right approach is in how you frame the statement. If you say that you've quit your job so you can start learning about the field, it does sound amateurish and hokey. However, if you frame this as a commitment to a new path in your life, it can come across very differently. You might say that recently you have decided to leave your job ("quit" is not necessary) to devote yourself full-time to deepening your knowledge base in your chosen field and applying for graduate programs, to maximize your success in that area. How you phrase this is up to you, but the important thing is for it to come across as a positive move of re-orientation and commitment, rather than "I just quit my job, and hope that grad school is the answer." Good luck!