I'm on desktop and would like to visit Quora as I visit it in mobile. Is there a way to do so? What URL should I use?
You can simply change your browser's User Agent to that of a mobile browser. Quora reads the User Agent when you load the page to determine which version of the page to show.
There are various tools to help with this (Chrome or Firefox add-ons, for example), or you can use the browser's built-in features to set it manually.
Once you set your User Agent and reload the page, you should see the mobile version of the page loaded instead of the desktop equivalent.