Is there a way to get Quake 1 2 and/or 3 to run at a modern resolution, namely 1920x1080?
I suppose the ideal solution would be to also incorporate some pillarboxing so that I can play at the intended aspect ratio, but honestly I really just want to play the game without all of my windows getting resized and moved around.

I understand there are some source ports that enable such features, but I am interested in making such changes to the version that ships with GOG.

As for Quake 1, in theory QuakeGL should support any resolution, which can be set using the command line parameters -height and -width. In practice I don't know if that works at all, at least on my end it crashed, and besides that you don't want to use QuakeGL as it is graphically inferior to most other versions out there.

I only got Quake 1 and 2 to run at 1920x1080 flawlessly and looking close to identical to the vanilla versions using source ports, namely Fitzquake Mark V for Quake 1 and Yamagi for Quake 2.

For Quake II there is an awesome mod which refreshes the whole graphic set.

https://www.youtube.com/watch?v=xiV0i6t2bQY

Quake2xp was released at the Quake Expo 2016 and can be downloaded from:
http://quake2xp.sourceforge.net/
and
http://www.moddb.com/mods/quake-2-xp

Here a comparison of the original with the engine conversion: https://www.youtube.com/watch?v=0uurJ-zjhok



For Quake 1 there are as well other engines avaible, here something from a setup guide for quake:
https://www.quaddicted.com/quake/recommended_engines