If it is intended to mate with other LEGO products, it is ABS. Just tweaked to be more elastic. Any discussion of materials published by LEGO always says LEGO and Duplo are ABS, with some SBS for tires and rubbery things.
There is some sort of polycarbonate used for the clear pieces, but these are rare, and they don't mate well with each-other, so LEGO actually tries to limit these in production kits.
I can't find a specific statement to this effect, but Quatro is almost certainly some tweaked ABS recipe, if only because they have to mate with other ABS pieces, and the tolerances are very tight for LEGO. Mixing plastics would be tricky (as it is for the PC pieces). Additionally, getting the colours just right so they match the colours in the other sets would be a nightmare if it was anything other than ABS.