  In cuprate superconductors, superconductivity appears below the CDW
transition temperature $T_{CDW}$. However, many-body electronic states under
the CDW order are still far from understood. Here, we study the development of
the spin fluctuations under the presence of $d$-wave bond order (BO) with
wavevector $q=(\pi/2,0),(0,\pi/2)$, which is derived from the paramagnon
interference mechanism in recent theoretical studies. Based on the $4 \times 1$
and $4 \times 4$ cluster Hubbard models, the feedback effects between spin
susceptibility and self-energy are calculated self-consistently by using the
fluctuation-exchange (FLEX) approximation. It is found that the $d$-wave BO
leads to a sizable suppression of the nuclear magnetic relaxation rate $1/T_1$.
In contrast, the reduction in $T_c$ is small, since the static susceptibility
$\chi^s(Q_s)$ is affected by the BO just slightly. It is verified that the
$d$-wave BO scenario is consistent with the experimental electronic properties
below $T_{CDW}$.
