Do I still need to fasten quarter inch drywall or can I just use adhesive?
Are modern construction adhesives sufficient to adhere 1/4&quot; drywall over solid plaster walls?

No
It's really good to add some adhesive to your drywall-over-drywall, but (speaking from experience) you should identify the studs before-hand and then drive nails or screws in. The reason is that the adhesive is coming out in a 1/4&quot; to 1/2&quot; bead. So even if you spread the adhesive liberally, it's only going to adhere to a small portion of the surface. Worse, you're not guaranteed to get the new drywall to &quot;suck up&quot; to the old one, meaning you might be left with a gap here and there, where the adhesive cured in a bead instead of being pressed flat. Against a 1/4&quot; board, that could leave a weak spot over a hole you can't see until something &quot;bumps&quot; it and breaks it.
I had to do this in my hall bath recently. I adhesived, mounted, and then nailed through the new and old into a stud. I didn't drive a lot of nails, but after doing so the first time, you could hear the adhesive being pressed between the boards (i.e. bubbles popping for a moment or two). The wall has not moved since I mounted it, and drove maybe half of what I would use on drywall straight over studs. Do it right, or do it a second time.