No infinite family exists. In fact all graphs with diameter $d$ and girth $2d+1$ have to be regular, and thus are Moore graphs. This was proved in
R. Singleton, "There is no irregular Moore graph", Amer. Math. Monthly 75 (1968), 42–43
See also the textbook "Algebraic Graph Theory" by Godsil and Royle (p. 90). It is unknown whether the Hoffman-Singleton graph is the largest Moore graph of girth 5 but if there are more, they have to be of degree 57 and of order 3250, thus there can only be finitely many more.