To find a counterexample, we should choose $C$ to have a lot of fibrant objects, but few cofibrant objects. So let $D$ be a combinatorial model category and let $C$ be the model category structure on the underlying category of $D$ in which every morphism is an acyclic fibration. This is again a combinatorial model category (with empty sets of generating (acyclic) cofibrations), and the identity functor $C \to D$ is a left Quillen functor. In $C$ every object is fibrant, but only the initial object is cofibrant.
In this case the question becomes: if the initial object of $D$ is fibrant, then is every object of $D$ fibrant? And of course there are many counterexamples.