I don't know where you live, and maybe the situation is different there. But speaking from a U.S. perspective:
One: Having a short gap -- a few months -- in your job history is not a big deal. Employers know that people sometimes lose or quit a job and don't necessarily get a new job immediately. If the gap drags out into six months or a year, then it becomes more of a problem. Employers start to wonder if there is a good reason why you can't get a job, or if you are lazy and didn't want to work.
Two: I have never had a potential employer ask for pay slips or any other proof of previous employment. But if that's common where you live, as others have said, I definitely would not lie about it and try to fake it. Ethical considerations aside, if you are caught in a lie in an interview, you are very unlikely to be hired. And if after you are hired they find out you lied during the interview, that is often grounds for being immediately fired. Lying during the interview means that you will spend your entire time with that company in fear that they'll find out. Don't do it.
Three: You don't have to tell us on this forum why you quit your last job, but an employer will expect something more than "for certain reasons". Frankly, I think that refusing to say why you quit would be the absolute worst possible answer. If I was the interviewer, I would take it for granted that if you had a good reason, you would have told me, so it must be a bad reason. And then I will imagine the worst possible reasons. Unless your reason is really bad, like you were caught stealing from the company or you beat up the boss, you are better off to tell the real reason. You may be able to paint it in a way that makes it sound okay or at least not so bad. Again, don't lie about it, but you can present things in a way favorable to yourself. If there's no way to make it sound good, then say, "Yes, I made a stupid mistake doing that and I will never do that again." Most companies are happy to have someone who learns from his mistakes. What they don't want is someone who makes the same mistakes over and over.