(Note that this will make your character model look weird in menus but the actual game will look totally normal).

Step 1
Go your Quake 3 installation folder (usually C:\Program Files (x86)\Steam\steamapps\common\Quake 3 Arena). Weirdly, Quake III Arena and Quake III Team Arena have different config files, despite being located in the same Steam installation folder, so instead of opening "baseq3" like you would to find the Quake III Arena config file, open the “missionpack” folder. In there, open the file named “q3config.cfg” with Notepad and press CTRL+F. In the search box search for “customheight” and press enter and you will find 2 lines:
seta r_customheight "1440"
seta r_customwidth "2560"
Change the values in the quotes after these lines to match your screen’s resolution. Mine is 2560x1440 so that is what I changed them to (as seen above).

Step 2
Just below these lines you will find another that says:
seta r_mode "3"
Make sure to change this value to “-1” so that the game will use the resolution you put in above.

Step 3
Make sure to press File→Save in order to save these changes.

In order to make the FOV look normal at your desired resolution you may need to change it.

Step 1
In the “q3config.cfg” file, CTRL+F and search for “cg_fov” to find the line:
seta cg_fov "110"
Change the value to change the FOV. I find 110 looks best to me personally.

Step 2
Make sure to press File→Save in order to save these changes.

Step 1
In the “q3config.cfg” file, CTRL+F and search for “maxfps” to find the line:
seta com_maxfps "166"
Change the value to change your maximum FPS. Since I use a 165hz monitor, I set it to 166 (1 value higher than my monitor's refresh rate) to ensure I am always getting the maximum amount of frames my monitor can display. That way, the game looks as smooth as possible.

Step 2
Make sure to press File→Save in order to save these changes.

Hope this helps! :D

I also recommend checking out my other guide on how to get Upscaled Textures, an HD Main Menu/HUD, and HQ Sounds: https://gtm.steamproxy.vip/sharedfiles/filedetails/?id=2960682722