First off, by the isomorphism (correspondence) theorem, every prime ideal of $\mathbb Z[x]/(9x^2+1)$ corresponds to a prime ideal of Z[x] which contains the ideal $(9x^2+1)$. Now $(9x^2+1)$ is clearly a prime ideal since $\mathbb Z[x]$ is a UFD and in a UFD irreducible elements are prime (hence the ideal generated by the prime $9x^2+1$ is a prime ideal). So the ideal $(9x^2+1)$ is a prime ideal of $\mathbb Z[x]$ which contains the ideal $(9x^2+1)$. But this ideal is not maximal using the fact that principal ideals in $\mathbb Z[x]$ are not maximal ideals. Hence we have found a prime ideal of $\mathbb Z[x]$ which contains $(9x^2+1)$ and is not maximal, so then $\mathbb Z[x]/(9x^2+1)$ also has a prime ideal which is not maximal.
Could someone confirm or point out a mistake in my solution? Thank you.
You are entirely right. And the non-maximal prime ideal in $\mathbb Z[x]/(9x^2+1)$, that corresponds to $(9x^2+1)\subseteq \Bbb Z[x]$, is the zero ideal.