Following ideas that go back to Cannon, we show the rationality of various generating functions of growth sequences counting embeddings of convex subgraphs in locally-finite, vertex-transitive graphs with the (relative) falsification by fellow traveler property (fftp). In particular, we recover results of Cannon, of Epstein, Iano-Fletcher and Zwick, and of Calegari and Fujiwara. One of our applications concerns Schreier coset graphs of hyperbolic groups relative to quasi-convex subgroups, we show that these graphs have rational growth, the falsification by fellow traveler property, and the existence of a lower bound for the growth rate independent of the finite generating set and the infinite index quasi-convex subgroup.