This is motivated by a computer-generated conjecture that bipartite distance-regular graphs are hamiltonian. I decided to check the case of Moore graphs first.
The cycles and complete bipartite graphs are hamiltonian (trivial).
The girth-6 graphs are the incidence graphs of the projective planes. The classical finite-field planes are hamiltonian by Singer, but there are lots of nonclassical planes.
And what about the girth-8 and girth-12 case?
EDIT: Corrected the girth-6 case as pointed out by Gordon Royle.