I installed Quake today, and got the same error message (wglCreateContect failed) when attempting to use the default GLQuake shortcut. Winquake however, did work ... but blocky and pixel looking.
I eventually solved it by renaming the opengl32.dll file so it doesn't get used.
Then I had to create a Brightness scheme for my monitor, as my standard setting is way too dark for GL Quake, and in-game settings don't make any difference.
Now everything looks & works great.
UPDATE - Alas it did not remain so, as I have an issue on occasion where GL_Exec commands are visible on screen overlaying everything ... like a ghosting overlay. I can mostly use ALT-TAB to get rid of it, but that gives the screen (window) a bright lime green border that pulses and irritates. So it is best to restart the game ... sometimes a few times until it shows a normal Quake start screen.
Windows XP Laptop (32 bit).
EDIT I decided to try some modern Quake engines, to see if things were improved.
First up was
Fitzquake, which worked very well, but does not support the non CD music files mentioned here.
So next up I tried QuakeSpasm, which is a fork of Fitzquake. It was almost as good (what's with the coming and going bright Lime green window border?), but did support the non CD music files, so I am mostly happy with it.
NOTE - It seems to me that the GL_exec code I see with standard GLquake, that has the issue I mentioned above, is what is described as GL initialization, and for some reason the Quake menu/demo is starting too early, before that is cleared, and then not clearing it.