I used to use those to wipe glaze off the bottoms of my pottery before firing. I just hosed them off--they cleaned up pretty easily. Poop is different of course, but it might be worth trying. You'd definitely need to let them dry well before using again, but the ones I have dry pretty quickly.
I think a porous material of some kind could work if it was machine washable. My concern with carpet tiles would be how stiff they are. You can't really throw them in the wash to sanitize them. I suppose you could try to soak and scrub them in the sink but that seems like a lot of work when there are other good options.
Admittedly I am approaching this more as a reptile owner than a quail owner. I have no experience with quail lol. But I do use felt as the substrate for my leopard gecko. Porous substrates like reptile carpet (fancy felt) are pretty common in the reptile world, but you still have to be really careful about the potential hygiene issues. Reptiles can also get coccidiosis and other diseases if their tank is not properly cleaned and maintained. I have several identical pieces of felt for the bottom of Hoshi's tank and I have to change and thoroughly launder them regularly. And he only poops a few times a week! I can only imagine what kind of biohazards quail could produce