Hello!
Unfortunately, non-assistance animals are not permitted as pets in University residences.
From my experience of living in Glasgow, I would recommend looking at private options. If you would like to stay near the university, I would suggest looking at the West End (including Hillhead, Partick, Finnieston, etc.). If you are happy to get the subway to campus, it may expand your options. I have known people to keep birds as pets in Glasgow, but I am unsure of how they sorted that.
Finding a flatmate on something like Spare Room may allow you to lodge with somebody who owns the flat, which may be more relaxed than going through estate agents, etc.
Best of luck!
Ben, 3rd Year
University of Glasgow