Would it be possible to see a galaxy like this with night vision glasses?

You would need something like Generation 4 night vision in order to do that.

And those are not cheap.

https://www.nightvision4less.com/gen-4-night-visio...

A better and cheaper solution would be large aperture binoculars. And a good dark sky site.

https://www.telescope.com/Orion/Binoculars/Astrono...

Night vision goggles work by converting a near-infrared image into a visible image.  M31 is not much brighter in the near-infrared than it is at visual wavelengths, so I'd say "no", in answer to your question.

Night vision is just light amplification, team that up with a telescope, you may see a galaxy better.

Astronomical cameras have a wider spectrum of light than the human eye, amazing what they can see. Better than any night vision.

My good telescope (11 inch) with low power widefield eyepiece, you can see Andromeda about like your image. Figure 20* across field of view.

Good viewing conditions you can see much of it with the naked eye.

Just that you cant see the dimmer stars.

that depends on the amount of pollution (dust) in the air and the amount of active lightsources in the area

I'm not sure what you mean by "night vision glasses." If you're talking about regular all-purpose night vision glasses that cost under a hundred dollars that work by filtering certain colors of light, no. If you're talking about astronomy night vision glasses or goggles designed for stargazing that actually amplify and clarify light, yes, but the most inexpensive of those start at around $1,700 and go up to about $12,500. 

You will not be able to duplicate what you see in that image above because that image has been graphically manipulated to get rid of every point of light in the night sky, which would be amplified as much as Andromeda. You'd see a whole sea of stars, so much so that Andromeda would appear less dramatically. That's why the image has been edited to eliminate all other points of light-- to make Andromeda stand out like that against a solid black/dark background instead of being surrounded by other points of light. 

In fact, it's very possible that that image is not even one picture but two images superimposed. One of Andromeda taken when night was at its darkest and another of the moon low on the horizon and the sun not too far below the horizon. Then someone merged the images, the upper right corner being one picture and the lower left corner including the entire bottom until the clouds stop another, and then using an app to smooth the two images seamlessly together.

you could try it and see and then you'll know