Coloured FR4 can be made, but it is a special order with significant leadtime and minimum order qty.
AFAIK there are no differences in the process, just a case of the PCB house sourcing the material.
You won't find a PCB supplier who offers it as standard unless they have some leftover from a previous job, and as it's an unusual requirement this is pretty unlikely
For small qtys you might find an exotic RF material that may be close to the colour you want, but chances are you'll still have a minimum order qty.
If you only want a few square feet, forget it. Not going to happen.
If you want a starting point, this company does non-copperclad FR4 in a few colours
https://www.gaatech.com/fr4-g10-sheet-laminates-epoxyglass/