No, Quake 3 for Linux runs in Linux without any kind of emulator.
It doesn't matter which version you buy, Quake 3 for Linux only needs the data files (pk3's) which are the same throughout Mac, Windows and Linux. I have the Mac version of RTCW and it works in Linux.
That is just a Linux version that comes with the installer on it. You don't need that if you already have the Windows version.
The same thing goes for RTCW, Quake 4, Doom 3, Doom 3 expansion and Quake 2, etc. UT2k4's Windows disc has the Linux installer on it.
So you just
sh /path/to/linuxinstaller.bin and a graphical installer will run just as if it were in Windows.
If you need any help I'm all about games in Linux, but I will have to go from memory right now, because I don't have Linux and can't have Linux on my desktop until I can figure out how to use a Gentoo LiveCD and install with debootstrap.
It sounds really complicated so it may be a while. :S