I've been trying to run WinQuake on Windows 10 and have run into a problem:

The game runs, but the screen isn't centered; it's translated upward a bit, leaving a black strip at the bottom of the screen, and cutting off a portion of the top of what should be shown.

I was wondering if there was any way to fix this?

Obvious question that's going to come up is: Why am I running WinQuake? Well, it was the only version of Quake that seems to run the Alien Quake mod without scrambling some of the graphics. It's not the mod - I tried running WinQuake both without, and with the mod, and the same dislocation of the screen occurs.

Alien Quake as in this mod? http://www.moddb.com/mods/alien-quake
Only I've just been able to play that using Quakespasm without any issues.

Got to say, though, I'm not a fan of the mod itself. The maps I played were short and enclosed meaning you regularly bump into architecture and objects, and they are so very dark. Yeah, I get the dark and enclosed spaces is because of the Aliens theme but I've seen it done much better. Knowing what to do and where to go is also not clear. Doors often look like walls, too, so I missed some a few times and kept running back and forth. Also, the mod is buggy as I use the mousewheel to switch weapons and the mod doesn't react to mousewheel up, only down, and it switches from the grenade launcher to the pulse rifle after firing a grenade (same gun just different mode as shown by the red/yellow light on the gun). Plus, even on easy, the damage you take is huge.

Anyway, in regards to your issue, I'd be looking at graphics drivers first, being as you say you are getting scrambled graphics with this mod (except in WinQuake, so I'm thinking maybe it is the hardware graphcs acceleration that is giving the scrambled graphics and that software mode is unaffected) and then with WinQuake you are getting a displaced screen, so it could be something to do with the screen scaling for the screen resolution you are using.

I don't know about WinQuake specifically, but I ran into this issue with another game recently in Windows 10. The solution was to set scaling in the Display Settings for Windows to 100%. It usually defaults to higher like 150% or so. This shifted one or two of my games somewhat diagonally off the screen. I can't remember which game( s ) but hopefully this will help you.

I managed to resolve this problem by scaling the screen - just playing around with options caused the screen to shift over a smidgen.

WinQuake has limited max resolution which was limited by the technology at the time; I think those were the DirectX 7 days. You'll need a newer sourceport for modern resolutions. What you're seeing might be your graphics card trying to stretch a lower resolution up to a higher native one.