 Ext groups between polynomial-ring residue modules

When $X$ acts as zero, resolve the first module with the Koszul resolution on $(p,X)$. Writing $N=\mathbb{Z}/q$, the resulting groups are $\ker (p:N\to N)$ in degree zero, $(N/pN)\oplus \ker p$ in degree one, and $N/pN$ in degree two, with zero higher groups. This works for composite as well as prime moduli.