Could duragesic show up as morephine or hydromorphone in a drug test?

No it won't show up as morphine or hydromorphone. Its totally unrelated chemically. If there is a test category for 'opioids' this might test for fentanyl, but I believe this is fairly uncommon. If there is a test for 'fentanyl' or 'synthetic opioids' these will/may test for it, respectively.