1. I have a talking dog.

2. I have a dog that is talking.
3. I have a dog which is talking.
4. I have a dog who is talking.
[These are example sentences. #1 can be changed into the others. Can we use 'who' as in #4 when the antecedent is an animal?]

"Who" or "which" is really a matter of personal preference, depending on how you perceive animals. One might also say, "I have a dog that talks."