  We calculate the thermal Hall conductivity $\kappa_{xy}$ for the columnar
valence-bond solid phase of a two-dimensional frustrated antiferromagnet. In
particular, we consider the square lattice spin-$1/2$ $J_1$-$J_2$
antiferromagnetic Heisenberg model with an additional Dzyaloshinskii-Moriya
interaction between the spins and in the presence of an external magnetic
field. We concentrate on the intermediate parameter region of the $J_1$-$J_2$
model, where a quantum paramagnetic phase is stable, and consider a
Dzyaloshinskii-Moriya vector pattern associated with the couplings between the
spins in the CuO$_2$ planes of the YBCO compound. We describe the columnar
valence-bond solid phase within the bond-operator formalism, which allows us to
map the Heisenberg model into an effective interacting boson model written in
terms of triplet operators. The effective boson model is studied within the
harmonic approximation and the triplon excitation bands of the columnar
valence-bond solid phase is determined. We then calculate the Berry curvature
and the Chern numbers of the triplon excitation bands and, finally, determine
the thermal Hall conductivity due to triplons as a function of the temperature.
We find that the Dzyaloshinskii-Moriya interaction yields a finite Berry
curvature for the triplon bands, but the corresponding Chern numbers vanish.
Although the triplon excitations are topologically trivial, the thermal Hall
conductivity of the columnar valence-bond solid phase in the square lattice
antiferromagnet is finite at low temperatures. We comment on the relations of
our results with a {\sl no-go} condition for a thermal Hall effect previously
derived for ordered magnets.
