The ^{181}Ta quadrupole resonance [nuclear quadrupole resonance (NQR)] technique is utilized to investigate the microscopic magnetic properties of the Weyl semimetal TaP. We find 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, ν_{Q}=19.250  MHz, and an asymmetric parameter of the electric field gradient, η=0.423, are 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_{1}T) is measured for the f_{2} line (±5/2↔±3/2 transition). We find that there exist two regimes with quite different relaxation processes. Above T^{*}≈30  K, a pronounced (1/T_{1}T)∝T^{2} behavior is found, which is attributed to the magnetic excitations at the Weyl nodes with temperature-dependent orbital hyperfine coupling. Below T^{*}, the relaxation is mainly governed by a Korringa process with 1/T_{1}T=const, 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.

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