It can. That does not necessarily mean it will.
A background check for employment is not a static thing that is always the same regardless. What shows up, or doesn't show up, in a background check depends entirely on how deep and thorough a check is done.
Is it likely to show up? Probably not. Can we carve it in stone that it will not? No.
Two things I am tired of typing: 1.) A wrongful termination does not mean that you were fired for something you didn't do; it means that you were fired for a reason prohibited by law. 2.) The above answer, whatever it is, assumes that no legally binding contract or CBA expressly says otherwise. If it does, the terms of the contract apply.