EDIT: I have been asked in a comment what a filtered quasi-isomorphism of DG-coalgebras is, so let me add an explanation of this to the answer. This notion does not presume a fixed filtration on a DG-coalgebra, but rather a class of admissible filtrations. We consider coaugmented DG-coalgebras, i.e., the basic field $k$ is embedded as a subcoalgebra into a DG-coalgebra $C$ and its image is annihilated by the differential. An increasing filtration $F$ on $C$ is admissible if $F_0C=0$, $F_1C=k$, the components $F_iC$ are preserved by the differential, and the filtration $F$ is compatible with the comultiplication.
Not every coalgebra admits an admissible filtration at all! A coalgebra is called conilpotent if it has an admissible filtration (just as a coalgebra, let us drop the condition of compatibility with the differential for a moment). In this case, it has a certain maximal admissible filtration (which is preserved by the differential automatically). However, filtered quasi-isomorphisms of conilpotent DG-coalgebras are defined in terms of arbitrary admissible filtrations. A morphism of DG-coalgebras is a filtered quasi-isomorphism if admissible filtrations can be chosen on its source and target so that the morphism is compatible with them and is a quasi-isomorphism on all filtration components.