Is it possible to finish PhD quicker than formally possible? I said to my supervisor that I would like to finish my PhD very quickly. He said that formally I can submit my thesis 2 years after starting my degree at earliest, but that I am doing very well and it should be possible.

I verified the institutional requirements and indeed I cannot submit my PhD thesis in less than 2 years. However, I would like to submit the thesis in 1 year and 3 months. But I would like to ask you if this could somehow be possible and if yes, then how and what I should do.

I could talk to my supervisor in person about this, but the problem is that he would find it hard to believe me that much and thus he would be reluctant to help me or what is even worse, I may hurt the relationship with my supervisor. I have been doing my doctoral degree for 9 months in mathematics and some big parts are coming together, yet still there is a lot of work.
I believe that I could finish it in 6 months, but my supervisor would believe me only if he saw everything completed. The problem is that by then it would be too late for me.

Edit: I study in the UK. If I can finish my PhD thesis earlier, and then move to a preferred place for a postdoc, I do not see a reason to delay. I need to know now if I can finish early, so that I could apply for the scholarships supporting my postdoctoral visit. The scholarships I am going to apply require that I start within certain dates and can be applied for only at specific times during the year twice or once a year.

I have fulfilled all the credit requirements already and meet my supervisor every week while formally it is required only once a month.

The specific rule is:

In some cases the graduate board may allow you to submit your thesis early, before the end of your standard period of study. A Postgraduate Research Tutor can only recommend this if:
(a) the thesis is complete and ready for submission and
(b) has been read by your supervisor in its entirety. Your supervisor must confirm in writing that they have read your thesis and that their view is that it is ready to be examined.

The earliest date you can submit your PhD thesis is at the end of 2 calendar years of full-time study.
I don't understand what you want; do you want us to contact your supervisor for you?  We can't find some magic exception in the rules for you.  Either you need to suck it up and accept this won't happen, or you need talk to your supervisor and convince them that this is a good idea.  

Let me just add that this sounds like a terrible idea to me.  UK Ph.D.'s are way too short anyways; rushing through one will just throw you out into the cold hard reality of the job market totally underprepared.  You don't seem to have thought about the "If" in "If I can finish my PhD thesis earlier, and then move to a preferred place for a postdoc, I do not see a reason to delay." or thought about what happens at the end of your postdoc.  If you are really finished with your thesis at the end of 15 months, then it's not as though you'll twiddle your thumbs for the next several months; you'll have an incredibly valuable chance to start some other projects, and get a bit of a head start before heading into your postdoc.  That's time you'll never get back, and which you want to strain your relationship with your advisor in your rush to throw away.