I will sketch some aspects of an interesting Gromov-Witten theory on weighted projective planes introduced by Gross, Pandharipande and Siebert. It admits a very special expansion in terms of tropical counts (called the tropical vertex), as well as a conjectural BPS structure. Then I will describe a refinement or "$q$-deformation" of the expansion using Block-Goettsche invariants, motivated by wall-crossing ideas. This leads naturally to a definition of a class of putative $q$-deformed BPS counts. We prove that this coincides with another natural $q$-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined (joint with S. A. Filippini).