Yes, a quantum computer could be simulated by a Turing machine, though this shouldn't be taken to imply that real-world quantum computers couldn't enjoy quantum advantage, i.e. a significant implementation advantage over real-world classical computers.
As a rule-of-thumb, if a human could manually describe or imagine how something ought to operate, that imagining can be implemented on a Turing machine. Quantum computers fall into this category.
At current, a big motivation for quantum computing is that qubits can exist in superpositions,$$
\left| \psi \right> = \alpha \left| 0 \right> + \beta \left| 1 \right>, \tag{1}
$$essentially allowing for massively parallel computation. Then there's quantum annealing and other little tricks that are basically analog computing tactics.
But, those benefits are about efficiency. In some cases, that efficiency is beyond astronomical, enabling stuff that wouldn't have been practical on classical hardware. This causes quantum computing to have major applications in cryptography and such.
However, quantum computing isn't currently motivated by a desire for things that we fundamentally couldn't do before. If a quantum computer can perform an operation, then a classical Turing machine could perform a simulation of a quantum computer performing that operation.
Randomness isn't a problem. I guess two big reasons:
Randomness can be more precisely captured by using distribution math anyway.
Randomness isn't a real "thing" to begin with; it's merely ignorance. And we can always produce ignorance.