I suppose anything can happen, but I am not sure what the nature of your question is related to. Is this after or before surgery?
It is hard to give you a definite answer. Usually injury does not make the nose droop but if the nose is broken and has changed shape there can be the illusion that it is drooping. The bottom half of the nose is cartilage and usually is pretty resistant to injury, unlike the bone on top. I think it is best to see an ear nose and throat doctor or facial plastic surgeon to determine whether it was broken and what your future options may be. Hope that helps!