This is REALLY confusing. Do I need to download a windows file? I am extremely lost on how to do this.

Why on earth would you want to play Quake on a touchscreen?

Well, the installers are windows executables, so yeah. I suppose that the easiest is that you install it on a windows PC, look for and transfer the game files, then configure this "dosbox turbo".

All I have is a phone. Duh.

And I have no idea how to do any of this. Detail please.

What gave you the impression that buying PC games for an Android phone was a feasible idea?

You'll need to get the .pak files from a Quake install somehow. You will likely need a Windows PC to do this for most digital distribution methods.

Once you have that, using an Android source port of Quake seems more sensible than using the DOS version in Dosbox. Something like this:
https://play.google.com/store/apps/details?id=com.n0n3m4.Q4A&hl=en

Did you bother to read the linked guide on the Google Play page?
https://sites.google.com/site/dosboxturbo/home/beginning-dosbox-turbo-for-newbies

Yes. Or two different guides actually. Very confused and have no pc.

Actually, that guide I hadn't seen. Looking now.

Well, that is for a dos game, not a windows, and still requires a pc.

What should I search for on google play when trying to find out how to extract a .exe file on android?

An exe isn't an archive to be extracted. Some are merely archives, but generally exe are executables that contain code. There is no set way to "extract" them since the compiled code within could be literally anything.

Don't we have Quake source port for android?
Google shows QI4A as one of them I haven't tried it though.

That thing doesn't look like quake. How would I use it? Is it safe to use?

This is why there needs to be anforum on this.

this is a general forum for PC GAMES
if you want help with android go to an ANDROID forum

..seems so simple

and you never answered why you seemed to think that buying pc games for a phone is a great idea

long story short : get a windows pc