  This Letter theoretically discusses quasiparticle states and nuclear spin
relaxation rates $T_1^{-1}$ in a quasi-one-dimensional superconductor
(TMTSF)$_2$PF$_6$ under a magnetic field applied parallel to the conduction
chains. We study the effects of Josephson-type vortices on $T_1^{-1}$ by
solving the Bogoliubov de Gennes equation for $p$-, $d$- or $f$-wave pairing
interactions. In the presence of line nodes in pairing functions, $T_1^{-1}$ is
proportional to $T$ in sufficiently low temperatures because quasiparticles
induced by vortices at the Fermi energy relax spins. We also try to identify
the pairing symmetry of (TMTSF)$_2$PF$_6$.
