Let $R$ be a commutative Noetherian ring, $I,~J$ be two ideals of $R$, and $M,~N$ be two $R$-modules. Let $S$ be a Serre subcategory of the category of $R$-modules. We introduce Serre cohomological dimension of $N, M$ with respect to $(I,J)$, as ${\rm cd}(I, J, N, M)=\sup\{i\in \Bbb N_0: H_{I,J}^{i}(N, M)\not\in S\}.$ We study some properties of ${\rm cd}(I, J, N, M)$, and we get some formulas and upper bounds for it.
