Desargues graph, also known as the generalized Petersen graph $G(10,3)$ has girth 6 and also contains cycles of length 8. There exist three 10-cages, smallest cubic graphs of girth 10. They have 70 vertices and none of them is vertex-transitive. For more information about the motivation
see here.

Therefore it is quite surprising that Gordon's example is so small. By the way, it is also arc-transitive. One may consider it as a Levi graph of a self-dual, flag-transitive 3-configuration of 19 points and 19 lines. Configuration contains triangles but has no quadrangles.

Here is a geometric realization of this graph.

Configuration with triangles but no quadrangles

A natural question is, whether this is the smallest graph with required properties.

Another related question is, whether there exist smaller bipartite graphs if the vertex-transitivity condition is dropped.