Is it possible to play the Steam version of Quake on Ubuntu?

Yes, you can; this post on the Steam community forums is a comprehensive overview of Q1 engines.

I don't know about Q2 or Q3A, but as far as I know they work the same way; you can just install one of the open-source engines on top of the data files provided by Steam.

The same goes for the Steam versions of Doom 1/2 and Duke3d, by the way.

However, you cannot install Windows-only games using your Linux steam client. To make this work, you'll need to either have a Wine install of Steam or have Steam installed on Windows (virtual or real hardware) in order to get the data files in first place.