I'm not 100% sure, but I think this is very likely. Mark Hovey has recently studied model category structures on $Arr(\mathcal{C})$ where $\mathcal{C}$ is a bicomplete, closed, symmetric monoidal model category (e.g. the category of differential graded algebras). Some things can also be said with weaker conditions on $\mathcal{C}$. This is unpublished but he's been giving some talks on it and I imagine it will appear in the not-too-distant future. I'm not sure how much I should say without his permission, but you could always contact him directly. One of the points of his paper is that you can get lots of nice properties on $Arr(\mathcal{C})$. It's also possible others have studied this. Anyway, it turns out that the arrow category has a model category structure where the weak equivalences are squares

$\begin{matrix}M & \to & A \\\ f\downarrow & & \downarrow g\\\ M' & \to & A' \end{matrix}$

with $f,g$ weak equivalences in $\mathcal{C}$, and the cofibrations are squares as in your question with $f,g$ cofibrations in $\mathcal{C}$. This is the so-called injective model structure. There is also one where the weak equivalences and fibrations are squares with $f,g$ weak equivalences and fibrations respectively. This is the so-called projective model structure.

If you look into the paper *Algebras and Modules in Monoidal Model Categories* by Stephan Schwede and Brooke Shipley you'll see that if $\mathcal{C}$ satisfies something called the Monoid Axiom and is cofibrantly generated then you can get a model structure on $Mon(\mathcal{C})$ the monoids of $\mathcal{C}$. You can also get a model category structure on modules over a commutative monoid, and this seems to be the situation you're in. One thing which Hovey proves in his manuscript is that if $\mathcal{C}$ satisfies the monoid axiom (and the other hypotheses listed in the second sentence of this answer) then $Arr(\mathcal{C})$ under the injective model structure satisfies the monoid axiom.

Finally, the maps you're interested in--where the source is a dg module over the target (which is a cdga)--should also form a model category as the monoids in $Arr(\mathcal{C})$ under an appropriate product on $\mathcal{C}$. It turns out the correct product to work with is the pushout product, which can be found in Hovey's book *Model Categories*, but which I'll reproduce here:

For $f:X_0\rightarrow X_1$ and $g:Y_0\rightarrow Y_1$

$f \Box g : (X_0 \otimes Y_1) \coprod_{X_0\otimes Y_0}(X_1\otimes Y_0) \rightarrow X_1\otimes Y_1$

I'm really not comfortable saying more here, but please feel free to email me if you'd like some more details.