Yes, but you need to spell it as a separate word, and probably with a question mark at the end: "Köpek bizim ki?"
This sort of sentence is usually preceded by "iyi de": "İyi de, köpek bizim ki?" ("Okay but the dog is (already) ours, so what's the problem?" or something along those lines.)
I have a theory which I'll try to clarify with two examples:
You're somewhere where there are a lot of dogs and you want to show someone yours; Köpek bizimki: this dog is the one that is ours.
Your dog has run off and someone asks "whose dog is this?"; Köpek bizim: the dog is ours.
So with ki: more emphasis on the dog. Without ki: more emphasis on the possession.
It would be great if someone with more knowledge on this could confirm this though.