Please help!

Initializing OpenGL subsystem
...registered window class
...registered fake window class
...initializing QGL
...calling LoadLibrary( 'opengl32' ): succeeded
Couldn't find proc address for: wglGetExtensionsStringARB
Couldn't find proc address for: wglSwapIntervalEXT
Couldn't find proc address for: wglGetPixelFormatAttribivARB
Couldn't find proc address for: wglGetPixelFormatAttribfvARB
Couldn't find proc address for: wglChoosePixelFormatARB
Couldn't find proc address for: wglCreatePbufferARB
Couldn't find proc address for: wglGetPbufferDCARB
Couldn't find proc address for: wglReleasePbufferDCARB
Couldn't find proc address for: wglDestroyPbufferARB
Couldn't find proc address for: wglQueryPbufferARB
Couldn't find proc address for: wglBindTexImageARB
Couldn't find proc address for: wglReleaseTexImageARB
Couldn't find proc address for: wglSetPbufferAttribARB
...640x480 is unsupported in 32 bit
Shutting down OpenGL subsystem
...wglMakeCurrent( NULL, NULL ): failed
...shutting down QGL
...unloading OpenGL DLL
Initializing OpenGL subsystem
...initializing QGL
...calling LoadLibrary( 'opengl32' ): succeeded
Couldn't find proc address for: wglGetExtensionsStringARB
Couldn't find proc address for: wglSwapIntervalEXT
Couldn't find proc address for: wglGetPixelFormatAttribivARB
Couldn't find proc address for: wglGetPixelFormatAttribfvARB
Couldn't find proc address for: wglChoosePixelFormatARB
Couldn't find proc address for: wglCreatePbufferARB
Couldn't find proc address for: wglGetPbufferDCARB
Couldn't find proc address for: wglReleasePbufferDCARB
Couldn't find proc address for: wglDestroyPbufferARB
Couldn't find proc address for: wglQueryPbufferARB
Couldn't find proc address for: wglBindTexImageARB
Couldn't find proc address for: wglReleaseTexImageARB
Couldn't find proc address for: wglSetPbufferAttribARB
...640x480 is unsupported in 32 bit
Shutting down OpenGL subsystem
...wglMakeCurrent( NULL, NULL ): failed
...shutting down QGL
...unloading OpenGL DLL
Fatal Error: Unable to initialize OpenGL
--------------- BSE Shutdown ----------------
---------------------------------------------
idRenderSystem::Shutdown()
Unable to initialize OpenGL

Looks like it's trying to use a resolution that doesn't work.

https://www.gog.com/forum/quake_series/quake_iv_opengl_problems/post2

If there is an opengl32.dll in the installation folders, rename it. This will ensure the one from the graphics driver will be used.

Thanks for the suggestion. Alas, I had already seen that thread and tried it without success. Thanks for the suggestion. I searched but couldn't find any "opengl32.dll" file. Could it be under another name? Also, if so, what should I rename it?

opengl32.dll is the renderer. Older games often bring old versions with them and use that, instead of the one the graphics driver has.That can causes rendering problems.
It can be renamed to anything, so that the game can't load it.