VESA is just a standardization of video modes, it doesn't have anything to do with any sort of video acceleration.
Assuming you're running the original Quake, all of the rendering is done in software on the host CPU using the floating point unit (which is why Quake requires an FPU.) In order to get any sort of respectable frame rate at 640x480, you're going to need a very fast CPU.
My original gaming machine in 1997 with a Pentium 200 MMX would run the game adequately at 320x200, but would be a slide show at 640x480. I later upgraded it to an AMD K6/2 400 which made 640x480 just barely playable.
In the vintage gaming world, we usually call such things VDPs (video display processor) because they usually rely on the host CPU to do most functions. Some may have certain acceleration features like DMA, blitter, hardware sprites, etc. but are still referred to as VDPs.