  We report the $^{11}$B nuclear magnetic resonance (NMR) measurements on
non-centrosymmetric superconductors NbRh$_2$B$_2$ (superconducting transition
temperature $T_c$ = 7.8 K) and TaRh$_2$B$_2$ ($T_c$ = 5.9 K) with a chiral
crystal structure. The nuclear spin-lattice relaxation rate $1/T_1$ shows no
coherence peak below $T_{\rm c}$, which suggests unconventional nature of the
superconductivity. In the normal state, $1/T_1T$ increases with decreasing
temperature $T$ at low temperatures below $T$ = 200 K for TaRh$_2$B$_2$ and $T$
= 15 K for NbRh$_2$B$_2$, while the Knight shift remains constant. These
results suggest the presence of antiferromagnetic spin fluctuations in both
compounds. The stronger spin fluctuations in TaRh$_2$B$_2$ compared to
NbRh$_2$B$_2$ is discussed in the context of spin-orbit coupling.
