Quake 1 has difficulties being run under Windows XP and other modern operating systems. Since the release of Quake's source code, amateur programmers have modified the game so that it can be played on modern Windows operating systems, along with extra features such as better graphics. If you have XP, Vista, or Windows 7 and are having problems with Quake, follow these steps and it should be working in no time.
Quake on DOSBox also works in a modern computer. This option keeps the game in its raw version without improvements created by the Source ports. To install, simply open up the DOSbox configuration file. Mount the folder that contains the .exe file. Go to the folder by typing the letter drive you mounted it to, then type the name of the executable. Link to DOSBox
If you've purchased Quake via Steam, download GCFScape and look for quake.gcf in /steam/steamapps/ folder. Extract everything in it into a folder called "Quake", then use one of the options.