Usually, the derived category of some abelian category $A$ (I'm happy already with $A$-mod) is defined first taking chain complexes up to homotopy, and then localize at quasi-isomorphisms.

My question is, if one begins with $Chain(A)$=complexes in $A$ (instead of complexes up to homotopy equivalence) , and then localize at quasi-isomorphisms, do we get the same?

Denote

$$Chain(A)\overset{\tilde Q}{\to} [qis]^{-1}Chain(A)$$

the functor with the universal property w.r.t. this localization.

It is obvious that an homotopy equivalence is a quasi-isomorphism, but is it also obvious (or is it true?) that if $f\sim g$ are two homotopic maps then $Q(f)=Q(g)$? Is this category the same as the usual derived category?