Yes, the drug you're dealing with here is your run-of-the-mill Morphine.
Morphine is the prototypical, archetypical Opioid in every sense. Most other Opioids will be compared to Morphine in the course of defining their effects. I can tell you are not well-versed in this sort of stuff. Heroin, the well-known Opioid drug is just Morphine that has been treated with Acetic Acid/Acetic Anhydride as a sort of "cheap trick" to make the Morphine 1.5x-2x as potent.
All Opioids have nausea as a potential side effect. Morphine is known to cause relatively more nausea than than the Oxycodone you have mentioned. If it's at all possible, it would be worthwhile to request an alternative Opioid medication if the nausea is too much. For the time being, carbonated drinks, Cannabis, tea and OTC nausea aids can help improve your symptoms.
If you find yourself in the ER struggling with nausea they can possible give you some Ondansetron (Zofran) which is typically a very effective medication for nausea. It can also be prescribed and taken at home. I would ask if you continue having problems.