I suspect that $\#\mathbb{Z}[X]/(f,g)=|R(f,g)|$ holds for any two non-constant polynomials $f,g\in\mathbb{Z}[X]$, where $R(f,g)$ is the resultant of $f$ and $g$. I am however unable to prove it. I'd like to know whether or not this is true, and if so, a hint as to how to prove it.
For $f,g\in\mathbb{Z}[X]$ splitting as $f=a\prod_{i=0}^m(X-\alpha_i)$ and $g=b\prod_{j=0}^n(X-\beta_j)$ over $\overline{\mathbb{Q}}$, I understand the resultant of $f$ and $g$ to be $R(f,g):=a^nb^m\prod_{i=0}^m\prod_{j=0}^n(\alpha_i-\beta_j).$
Edit: From the first few replies it is clear that I rushed this post. I should mention that for my own purposes I only require a proof for distinct $f$ and $g$, both monic and irreducible over $\mathbb{R}$. More explicitly; $f$ and $g$ are both either linear, or quadratic with negative discriminant.
It seemed to me that the desired result would generalise, though clearly some assumptions must be made. Perhaps it is enough to assume both $f$ and $g$ monic, and the $\alpha_i$ and $\beta_j$ all distinct?