Are the model structure(s) on chain complexes and the triangulated structure interchangeable, or complementary?

I am somewhat new to homotopy theory and homological algebra, so I apologize if this is a stupid question.

I am wondering if the triangulated structure on the category $\mathrm{Ch}(\mathsf{A})$ of chain complexes in an abelian category $\mathsf{A}$ can be entirely replaced by a choice of Quillen model structure on $\mathrm{Ch}(\mathsf{A})$ and/or its relevant subcategories (bounded complexes above/below, etc.).

For example, suppose I knew only abstract homotopy theory (Quillen model categories etc.) and the model structures on chain complexes of sheaves. Would I be able to compute sheaf cohomology and the usual derived functors using only that data? Or is it still necessary to know the distinguished triangles?