The normal question about this topic would be: Can a classical computer implement a quantum computer? But now I wonder the exact opposite: can a quantum computer implement a classical one? Can we say that classical computation is a subset of quantum computing?
Yes, it is perfectly possible to run a quantum computer as a classical one, simply by keeping all the qubits in 0 or 1 and deciding not to use the capability to place them on superpositions of those states or entangle them to each other.
To be a bit more precise, the computation complexity class BQP (of problems that can be efficiently solved in a quantum computer) contains as a subset the class BPP (of problems that can be efficiently solved in a classical computer, if you allow for a finite probability of success) which itself contains the class P (of problems that can be efficiently solved in a classical computer). The Wikipedia pages for those complexity classes have more detail on those inclusions.