Could a quantum computer calculate the values of the Riemann zeta that are currently out of reach with classical computers? Could a quantum computer calculate the values of the Riemann zeta function that are currently out of reach with the aid of classical computers?
Any counterexamples to the RH would be somewhere in the range that we couldn't calculate the value in a neighborhood of the region at any time to get the result. Is there any way a quantum computer could do this and report back the value thru some sort of measurement process?
 A: Yes, a quantum computer can evaluate Riemann zeta function polyLog time as shown in a recent paper titled ‘Evaluation of exponential sums and Riemann zeta function on quantum computer’
https://arxiv.org/abs/2002.11094
A: 
Could a quantum computer calculate the values of the Riemann zeta
  function that are currently out of reach with classical computers?

This is generally true of any computer that has more processing capacity than existing computers whether it is a quantum computer or not.

Any counterexamples to the RH would be somewhere in the range that we
  couldn't calculate the value

True.

in a neighborhood of the region in any time to get the result.

Not true. The Riemann zeta function extends to infinity and an exception could be at any point arbitrarily beyond our ability to calculate it numerically. 
Some patterns that hold true for a very long time numerically, ultimately fail. Professor John Baez has some notable examples at this blog post. For example, he discusses one that fails at approximately $1.397*10^{316}$. 
By comparison, the Universe is about $4.32*10^{17}$ seconds old, and the fastest process in the Standard Model is the decay of the W and Z bosons which have a mean lifetime of about $3*10^{-25}$ seconds. So, at a rate of one computation per W boson decay for the lifetime of the universe, you would get about $10^{42}$ calculations, in a Universe that has fewer than $10^{90}$ particles in it (including neutrinos and dark matter particles, if they exist). This wouldn't come even close to finding the exception to the series identified by Professor Baez. Your quantum computer would need to do $10^{194}$ calculations per W boson decay time period, per particle in the Universe, for the life of the Universe, to test all of the possibilities numerically. Of course, if you can't time travel, you have far fewer seconds available to you to do that calculation than the age of the Universe, and you can't actually include every single particle in the universe in your quantum computer.
But, even that rate of calculation would be no guarantee of a result that would definitively find or rule out an RH exception.
In short, this problem would be impossible to be certain that you could solve via brute force numerical methods of any kind.

Is there any way a quantum computer could do this and report back the
  value thru some sort of measurement process?

It can search all sorts of numbers, but it can't check all of them in an infinite series. You might get lucky, or you might not.
This isn't to say that a quantum computer couldn't be useful. 
For example, it could use machine learning to look for near misses for counterexamples and a pattern might develop that would help you decide where it would be fruitful to look numerically.
