Diode vs quantum computer
Summary: an ounce of math wins over computers, quantum or not.

For most natural distributions of A (including samplings from a physical process like diode noise, not too much affected by conversion to digital or feedback from the state of the generator) and common rounding methods, we are able to predict future output given enough samples. A stream of zeroes is all it takes. With good likelihood that will start to match the generator's output, and keep on that way from there on, before $2^{32}$ steps, or just $2^6$ if A is rounded to integer before multiplication.

This follows from the fact that once and if $V_j$ reaches zero, it remains stuck there. And that if A is rounded to integer (either to nearest, lower, or higher) before multiplication, an iteration removes from $V_j$ as many bits of entropy as there are low-order bits at zero in the integer multiplicand that turns out to be used. This problem does not touch the original RANDU, because its multiplier is always odd.



Hereafter I consider using $V_{j+1}=65539\cdot(V_j+1)\bmod2^{31}$, which at least may avoid that issue.

Now the generator can be be fully predicted from future output, or not, depending on characteristics of A that are not explicitly stated in the question. It is entirely possible to have A such that after rounding it is always in a small set of integer values equal modulo $2^{31}$, and then A will have no influence on output.

However, for distributions of A arising from sampling a quantum phenomena similar to diode noise and not too much affected by conversion to digital or feedback from the state of the generator, the output will indeed be significantly influenced by A. Output will not be fully predictable from past output, because the multiplier is likely to vary unpredictably (for reasons explained in that other answer, for diode noise). I do not believe (or see anyone believing) that a quantum computer without a direct physical link to the particular source used for A is able to change that; at best it could improve to some degree on the advantage obtainable by a classical computer, or turn something almost predictable to totally predictable.

With the hypothesis in the previous paragraph, the generator will remain utterly distinguishable from random; but if we believe in the hypothesis of quantum physics to any degree (as overwhelmingly most physicians and engineers do), there will be some exploitable entropy in its output, so that seeding enough of that into a sound and generously dimensioned CSPRNG will yield a RNG computationally indistinguishable from random, including by the quantum computer in the question.