Playing Quake 1/2 on Linux?

You can play Quake 1 using the DOSBox emulator (it's compatible). That's probably the closest you will come to the original experience, compared to the source ports.

DOSBox emulation usually works really well, particularly for the more popular titles, which get a lot of testing. And any even remotely modern PC should have the performance to emulate it easily. You'll probably even be able to do multiplayer (DOSBox has IPX/SPX emulation).

Quakespasm is a great fork of Fitzquake, the best (faithful) Singleplayer engine there is: http://quakespasm.sourceforge.net/download.htm

If you cannot use a OpenGL engine, maybe give Tyrquake a try.

Enjoy!