For copper, at the anode, oxidation takes place. Since the electrode is not inert, the electrode will react. So copper is oxidised. Then at the cathode, it is reduction, so copper is reduced. There is nothing to do with H+: Hydrogen is more reactive than copper so copper will be preferentially discharged anyway.
However, if the metal is more reactive than Hydrogen, (for example iron (II), then yes hydrogen will be liberated, if I'm not wrong.