# Subclass of problems of an NP hard problem

I have an NP hard problem $$P$$ that takes in arbitrary $$G = (V, E)$$ as input. I have another problem $$Q$$ that I want to show is NP hard, and this problem has arbitrary complete graphs $$G'$$ as input. Is this enough to say that $$Q$$ is also NP hard, since it is a subclass of the NP-hard problem with arbitrary inputs, or is this insufficient?