Simply put:
Is there a useful version of pointfree convergence spaces? If yes, what are resources dealing with this?
This seems like an obvious thing to ask, since there is a pointfree version of topology (I mean locales, not the mathematical area of "topology").
One objection that comes to mind is, that the definition very explicitely mentions points. I can think of a way to eliminate this problem: replace $x$ with its principal ultrafilter $\{A : x\in A\}$. So one possible approach to create a pointfree definition is to start with a poset $(P,\leq)$ with elements thought of as "filters" with a reflexive relation $\to $ on $P$, such that analogous versions of the other axioms of a convergence space hold. But this is just one idea.