  We investigate electric quadrupole hyperfine structure constant ($B$) results
in $^{135}$Ba$^+$ and $^{137}$Ba$^+$ and compare their ratios with the reported
measurements of the ratio between the nuclear quadrupole moment ($Q$) values of
these isotopes. We carry out confidence tests for the reported experimental $B$
values and calculations of $B/Q$ from the present work. Inconsistencies in the
experimental $B$ values are observed in both the isotopes from different
experiments performed using the same techniques. The present calculations are
carried out using an all order relativistic many-body theory considering only
single and double excitations in the coupled-cluster ansatz. After a detailed
analysis of the results, the values of $Q$ we obtain for $^{135}$Ba and
$^{137}$Ba are 0.153(2)$b$ and 0.236(3)$b$, respectively, which differ by about
4% from the currently referred precise values.
