Almost every place to climb in WA is in a national park or reserve and dogs are prohibited.
Having said that, I have seen plenty of dogs at Churchies and Stathams and Mountain and I think that if you keep your dog under control, then no one will complain (except the ranger if he/she turns up). But if your dog runs about going crazy and barking, then keep it home as dogs that do that really piss off climbers and everyone else.
I have seen a dog fall from the top of MQ because the owners were not properly taking care of it – they were too busy climbing. Splattered dog is not nice. (But at least it stopped barking after that.)