No simple process for it.
first you crate the whole with hair particles. Then you convert the particles to a mesh, then you transfer wright paint to it by distance.
The color is not derived from the base mesh that way, but altered in engine with vertex paint.
if you unwrap the card mesh you can also run a custom paint texture on it - it is sometime useful.
To do that you need to keep the UV as it was for the hair to render correctly, and add a custom UV to the mesh which you use to filter in the base color.
As far as I’m aware, there is no way to actually sample the color and apply it to a mesh or a set of quads.
even if there was, unless you re-unwrap the mesh uv after editing the particle, every color change would affect all instances, defeating the purpose.