There is no inter-species transmission (as far as I know), so no, you can not get this version of the virus from your dog, cat, bird, lizard, etc.
HOWEVER, if someone with the virus sneezes on their dog, and that dog rubs their snot/spit on your dog, you can transmit active virus to yourself after petting your dog. I have no idea how long the virus stays viable in this chain. I rather suspect nobody else does, either. Probably best to not allow your dogs to play with other dogs for the duration.