Let $R$ be a Noetherian ring, $I$ and $J$ be two ideals of $R$, and $S$ be a Serre subcategory of the category of $R$-modules satisfying the condition $C_I$. We extend the notion of $S$-depth of $I$ on a finitely generated $R$-module $M$, denoted by $S-\depth(I, M)$, to the class of $ZD$-modules. Next, as a generalization of $S-\depth(I, M)$ and $\depth(I, J, M)$, the $S-\depth$ of $(I, J)$ on a $ZD$-module $M$ is defined as $S-\depth(I, J, M)=\inf\{S-\depth(\frak{a}, M): \frak{a}\in \tilde{\rm W}(I,J)\}$, and some properties of this concept are investigated. Also, the relations between $S-\depth(I, J, M)$ and $H^{i}_{I,J}(M)$ are studied.
