Thank you for your response. I also thought that it was a proper noun. My daughter is a homeschooler and is working her way through a grammar book and that sentence was part of her assignment. She had to go through all of the sentences and mark all the nouns and proper nouns. We both thought "Ralph's" was a proper noun, but I have the teacher/answer book and it is not marked as being a proper noun in the sentence. All I could think of was that it was being used as an adjective to describe whose paw was being shaken. I have no idea if that's actually correct, or if that means that it can't still be a proper noun? I didn't want to just gloss over it without actually giving her an explanation, but I didn't know!
"Officer Brady shook Ralph's paw and scratched his ears."
It's 100% definitely a proper noun, so your book is wrong, I'm afraid. The fact that it has an apostrophe to indicate possession has no bearing on things. And nor does the fact that it's the name of an animal (presumably a pet). In this case "Ralph's" is a proper noun in the 'genitive case' (genitive is just a posh word for possessive!). Here are some more genitives:
The dog's tail / The cat's paws / The lion's claws (Common nouns)