This problem is one occurred in Bertrand Toën’s Lectures on DG-categories Prop 4.3.4.
Let $M$ be a cofibrantly generated $C(k)-$model category. Then it is automatically a DG category and its underlying category is the model category. Denote by $Int(M)$ the full sub-dg-category consisting of all cofibrant objects of $M$. Typically we can (and will always) take $M$ to be the DG category of DG modules over a small DG category. Let $j:Int(M)\rightarrow M$ be the inclusion DG functor. Then Toën asserts that we can construct a DG-functor $$q:M\rightarrow Int(M)$$such that we have morphisms of DG functors $jq\rightarrow Id,qj\rightarrow Id$. moreover these morphisms are objectwise quasi isomorphism of DG modules.
As far as I can tell, such a functor can be defined on the underlying category, i.e. the abelian category of DG modules over a small DG category. I tried to use the small object argument to define the required DG functor but failed.
Best Answer
I think this is actually a mistake, and will only work over a field. In that case, every object of $C(k)$ is cofibrant. Lack and Rosicky actually prove that this is equivalent, under more basic assumptions satisfied in any $C(k)$, to the existence of such enriched cofibrant replacements in every cofibrantly generated $V$-model category. Their proof of necessity is in the appendix of "homotopy locally presentable enriched categories", while Shulman proved sufficiency in 24.2 of "homotopy limits and colimits and enriched homotopy theory." In general I think this statement needs a more laborious proof, directly verifying that the map from $Int(M)$ to the localization is homotopy fully faithful.