I’m afraid that doesn’t work. Possibly you are thinking of the expression: ‘turn your back on someone’, which suggests that you ignore someone or take no notice of someone.
Does the expression ‘to put smb’s back up’ originate from our pets? I would say that a cat put its’ back up literally when it doesn’t like another cat and is trying to frighten it. (Of course, there is the figurative sense in the test.)