(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) and open the “baseq3” folder. Then 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.

Note: If in-game, you can also open the console and do /cg_fov 110 or whatever number you want to play around with the FOV in-game.

Step 1
In the “q3config.cfg” file, CTRL+F and search for “maxfps” to find the line:
seta com_maxfps "0"
Change the value to change your maximum FPS. Setting it to 0 will unlock the fps, making it so it will be as high as your PC can output.

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

This step is optional but since you are already changing stuff, I highly recommend doing it.

Step 1
In the “q3config.cfg” file, CTRL+F and search for “lightning” to find the line:
seta cg_trueLightning "1"
Make sure to set the value to 1. This makes it so the beam from the Lightning Gun follows the crosshair a lot more closely (like it does in Quake III: Team Arena), making it much easier to aim than it is leaving this value at it's default setting.

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

Hope this helps! :D

If you also play Team Arena you'll need to do these steps for it's config file aswell:
https://gtm.you1.cn/sharedfiles/filedetails/?id=2959316453
I also recommend checking out my other guide on how to get Upscaled Textures, an HD Main Menu/HUD, and HQ Sounds: https://gtm.you1.cn/sharedfiles/filedetails/?id=2960682722