A functor which takes quasi-isomorphisms to isomorphisms factors through a triangulated functor? Let $\mathcal A$ be an abelian category. Let $\mathbb K(\mathcal A)$ be the homotopy category of complexes of $\mathcal A$. Now. consider a functor $F$ from $\mathbb K(\mathcal A)$ to $\mathbb E$, which is also a triangulated category. Suppose the $F$ takes  quasi-isomorphisms to isomorphisms. Now by the universal property of $\mathbb D(\mathcal A)$, the derived category of $\mathcal A$, we know there is a functor $F'$ from $\mathbb D(\mathcal A)$ to $\mathbb E$ induced by functor $F$.
Is $F'$ a triangulated functor?
By triangulated functor, I mean a functor which commutes with the translation functor and takes distinguished triangles to distinguished triangles.

If you don't assume that $F$ is a triangulated functor then this may not be true.
For example, let $F:\mathbb{K}(\mathcal{A})\to\mathbb{K}(\mathcal{A})$ be the obvious functor sending a complex to its degree zero homology considered as a complex concentrated in degree zero. Then $F'$ doesn't commute with the translation functor.
But if you intended to specify that $F$ is triangulated, then it is true, as explained in Pierre-Guy Plamondon's answer.
