. . .It's actually "Obsession" where Spock states that "my haemoglobin is based on copper, not iron". But if it really is haemoglobin, then it must have iron there, or it wouldn't be called haemoglobin. Either Spock is misusing words for the sake of his uncivilized friends, or then he's actually saying that the oxygen carriers in his blood have copper in addition to iron, perhaps boosting rather than replacing the function of that metal.
Some Earth organisms have the protein hemocyanin in their blood, which does use copper as an oxygen-binding agent. But they’re arthropods and molluscs. I don’t imagine hemocyanin would be very efficient for vertebrates.