  The $^{181}$Ta quadrupole resonance (NQR) technique has been utilized to
investigate the microscopic magnetic properties of the Weyl semi-metal TaP. We
found three zero-field NQR signals associated with the transition between the
quadrupole split levels for Ta with $I$=7/2 nuclear spin. A quadrupole coupling
constant, $\nu_{\mathrm{Q}}$ =19.250 MHz, and an asymmetric parameter of the
electric field gradient, $\eta$ = 0.423 were extracted, in good agreement with
band structure calculations. In order to examine the magnetic excitations, the
temperature dependence of the spin lattice relaxation rate
(1/$T_{\mathrm{1}}T$) has been measured for the $f_{\mathrm{2}}$-line ($\pm$5/2
$\leftrightarrow$ $\pm$3/2 transition). We found that there exists two regimes
with quite different relaxation processes. Above $T\text{*}$ $\approx$ 30 K, a
pronounced (1/$T_{\mathrm{1}}T$) $\propto$ $T^{2}$ behavior was found, which is
attributed to the magnetic excitations at the Weyl nodes with temperature
dependent orbital hyperfine coupling. Below $T\text{*}$, the relaxation is
mainly governed by Korringa process with 1/$T_{\mathrm{1}}T$ = constant,
accompanied by an additional $T^{-1/2}$ type dependence to fit our experimental
data. We show that Ta-NQR is a novel probe for the bulk Weyl fermions and their
excitations.
