  We apply $^{125}$Te nuclear magnetic resonance (NMR) spectroscopy to
investigate the Dirac semimetal ZrTe$_5$. With the NMR magnetic field parallel
to the $b$-axis, we observe significant quantum magnetic effects. These include
an abrupt drop at 150 K in spin-lattice relaxation rate. This corresponds to a
gap-opening transition in the Dirac carriers, likely indicating the onset of
excitonic pairing. Below 50 K, we see a more negative shift for the Te$_z$
bridging site indicating the repopulation of Dirac levels with spin polarized
carriers at these temperatures. This is the previously reported 3D quantum Hall
regime; however, we see no sign of a charge density wave as has been proposed.
