I'm having to write out lots of short exact sequences and commutative diagrams at the moment. There are standard notations for monomorphisms ($\rightarrowtail$) and epimorphisms ($\twoheadrightarrow$), and also for inclusions ($\hookrightarrow$). Yet the dual of an inclusion is a quotient map, and I am not aware of a standard symbol for this. Is there one?
Simply using $\twoheadrightarrow$ is not really enough, since it could refer to any epimorphism if unspecified. What is the most efficient way to indicate a quotient map?
PS: I am referring to inclusions/quotient maps more generally for quotients of sets, abelian groups, vector spaces, topological spaces etc.
I do not think that there is a commonly accepted notation. It also depends on the category whether it makes sense to speak about inclusions and quotient maps. It does in the categories of sets, groups, abelian groups, vector spaces, topological spaces, etc. These examples are "concrete categories" whose objects can intuitively be understood as sets with some additional structure. In such categories one can define "proper" subobjects and quotient objects whose underlying sets are subsets and sets of equivelence classes, respectively.
Anyway, you should create your own symbol for quotient maps, perhaps $$X \to_q Y .$$