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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?

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Distinguished triangles are uniquely determined by the model structure, they are precisely the homotopy cofiber sequences.

Furthermore, the triangulated structure is also uniquely determined by the model structure: the homotopy category of a stable model category is a triangulated category.

Since any practical example of a triangulated category is induced by a stable model category (a counterexample is a paper in Inventiones, by the way), there is no real reason to use triangulated categories; they are a lingering remnant of the time when all the powerful model-categorical tools discovered in the last 20 years were not yet available.

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    Awesome! Do you happen to know of a reference that discusses homological algebra from this point of view? – 2017-02-14
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    @ಠ_ಠ: The very basic aspects are treated in Chapter 7 of Hovey's Model Categories. Chapter 1 of Lurie's Higher Algebra treats more advanced aspects, albeit using quasicategories, which are, however, quite similar to model categories for many practical purposes. – 2017-02-14