No, a URL can only contain (a subset of) ASCII.
The browser is converting "აბცდ" into percentage-encoded entities for the actual URL that is sent to the server. In fact, you should be embedding it as percentage encoded string into your document to begin with, the browser is just covering that mistake for you.
What the browser shows in the address bar is something different. Modern browsers try to be as user friendly as possible and decode some percentage encoded characters to show in the address bar as human readable text. For anti-spoofing reasons, only some are decoded, not all. Georgian happens to be pretty safe, since it's hard to mistake it for any other similar looking characters.