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.