What does play the role of a subobject classifier for quotient objects? It is known that in the category of sets the dualization of the notion of a subobject classifiers does not work because the only object admitting a morphism into an initial object is the empty set.
But if we look at the idea of a subobject classifier (to index subobjects), then we can see that in the category of sets quotient objects (defined as equivalence classes of epimorphisms that start from a given object) have canonical representatives: those are equivalence relations on the given set. 
So, in the category of sets quotient objects of an object $a$ can be classified by the set $E(a)\subseteq \mathcal P(a\times a)$ of all equivalence relations on $a$. 

Question. Has the set $E(a)$ of equivalence relations some categorial counterpart, which would index all quotient objects of a given object?

 A: One way to write the universal property of this object $E(a)$ is as follows:
a map $x \to E(a)$ is the same as an isomorphism class of epimorphism $x \times a \twoheadrightarrow k$ in $Set/x$, that is a diagram
$$ x \times a \twoheadrightarrow k \to x$$
whose composite is the first projection.
So it does "feel like" a subobject classifier, but it is not the dual notion (the dual of a subobject classifer, would have a universal property specifying what are morphisms out of it, not into it).
For this universal property to make sense in a category $C$, you need pullback of epimorphism to exists, or if you restrict to some specific class of epimorphism, that the class of epimorphism under consideration is stable under pullback.
You can also consider the universal property, equivalent in set, but that might be different in more general category:
a map $x \to E(a)$ is the same an an (isomorphism class of) equivalence relation on $x \times a$ which is included in $\Delta_x \times a \times a$.
Where "equivalence relation" on $z$ means subobject of $z \times z$ satisfying the usual stability properties...
This version of the universal property makes sense more generally in any category with finite limits (pullback of monomorphism is enough actually). The two are equivalent in an exact category.
As pointed out by Andreas Blass in the comment, in an elementary topos, these two universal properties make sense and defines a universal objects.
