The unfolded protein response isn't related to the kind of protein that you are eating - so no, you don't need to be worrying about eating protein. Basically, your body is making proteins all the time, including enzymes, to do things in your body, and sometimes, for whatever reason, they don't get folded up properly, which is a Really Bad Thing. It's important for your cells to react to this, and get rid of those unfolded proteins, because they can be really damaging.
It's not connected to dietary protein.