My department offers full fellowships (tuition + stipend, healthcare is publicly funded in the country) to most academic Master's students. They are all limited to two years. The department does not have the ability to extend the scholarship, nor does the university.
That being said, it happens that some students take longer than 2 years to defend (it's an academic Master's, so it's generally expected that the students will have published/submitted at least one or two papers before they graduate). In those cases, the remaining months are fully funded by the advisor, usually in the form a research assistantship from a grant that is directly related to the student's Master's project.
However, if you were at my institution, your request for an extension would sound naive and borderline preposterous and arrogant. An extension is something that occurs organically as you show that you are doing good work and need the extra time to finish and polish your thesis. This is not something that is requested as soon as you start doing research. If your advisor/university is going to invest more money on you, you must demonstrate that you are worth the money and have a good motive not to have finished your thesis in time. Writing a "qualified" thesis is, I'm sorry, a terrible excuse.
The least awkward way to approach anyone about this is to ask about the possibility of (and not demand!) an extension once you are more acquainted with your advisor. A good way to approach them about this issue would be by leading a conversation like this:
As you know, I have been studying the effect of wheel geometry in the
performance of hamster-powered supercomputers. I demonstrated that
heart-shaped meshes are very efficient, but as you know, that's the
expected result, given that hamsters tend to work harder in the
presence of love-displaying figures. I think that it would be very
promising to investigate whether plastic or steel wheels are more
efficient. If we demonstrate that plastic wheels are just as good as
steel, we could see a significant cost reduction in hamster-powered
supercomputers, and would certainly yield a very impactful paper. But
as you know, that would require 3D printing the wheels, which would be
very time-consuming, and would likely require more time than the four
months I have left. I will start working on the plastic wheels as soon
as possible, but if more time is necessary to finish the experiments,
do you think it would be possible to grant me a fully funded extension
of a few more months so that I can finish the last experiments and,
hopefully, write the paper?
All of that is, of course, assuming that the possibility of getting a funded extension even exists.
