I have asked this question here (*), but there are no answer.

Let $q \in \mathbb N \cap [2,+\infty[$ and $P \in \mathbb Q[x]$ with $\forall k \in [0,\deg(P)] \cap \mathbb N, P(q^k) \in \mathbb Z$.

Is it true that $\forall k \in \mathbb N, P(q^k) \in \mathbb Z$ ?