Will quantum computers be able to pass the Turing test sooner than classical computers?
I think a better phrasing to this question is - will cloud access to a quantum computer enable a classical computer to pass the Turing test sooner than a classical computer with no such access?
A quantum computer "alone" is very unlikely to pass the Turing test earlier than the above mentioned device, since quantum computers have technical difficulties performing large-scale simple tasks (e.g. multiply two 2048-digit numbers). Our usage of quantum computers relies on their ability to perform very specific tasks that are out of reach for classical computers.
In order to pass the Turing test, the computer needs to be able to cover an extremely wide variety of tasks; there is no reason to assume a quantum computer will be better than a classical computer in such a wide variety.
So the question remains - will cloud access to a quantum computer help with passing the Turing test? Here we can only speculate as well. For example, one can guess that good machine learning capabilities are essential to passing the Turing test (since you need to be able to tell the difference between a picture of a cat and a picture of a dog), and we hope that quantum computers will help us with machine learning. But this is still only in the realm of speculation.
My personal guess: Yes, it will help. But only because we have many years ahead of us until a computer passes the Turing test, and I believe(/hope/dream) that quantum computing will mature by then.