Will it hurt zoysia grass if a dog or cat pees on zoysia grass?
Yes female dogs and young males that squat to pee will cause brown spots from urine salts. Male dogs usually don't pee in the middle of the grass, but lift their leg against a tree or pole.
Cats usually don't pee much volume so not as much as an issue. And if you put out a sand box or pile of sand the cat will much prefer to use that and not pee in the middle of the grass.
A motion sensor activated sprinkler can deter both.