Motivated by the string diagrammatic approach to undirected tracial quantum graphs by Musto et al. [J. Math. Phys. 59(8), 081706 (2018)], in the former part of this paper, we diagrammatically formulate directed nontracial quantum graphs by Brannan et al. [Commun. Math. Phys. 375(3), 1777 (2019)]. In the latter part, we supply a concrete classification of undirected reflexive quantum graphs on M2 and their quantum automorphism groups in both tracial and nontracial settings. We also obtain quantum isomorphisms between tracial quantum graphs on M2 and certain classical graphs, which reproves the monoidal equivalences between SO(3) and S4+ and O(2) and H2+.