  We report $^{125}$Te NMR measurements of the topological quantum material
ZrTe$_5$. Spin-lattice relaxation results, well-explained by a theoretical
model of Dirac electron systems, reveal that the topological characteristic of
ZrTe$_5$ is $T$-dependent, changing from weak topological insulator to strong
topological insulator as temperature increases. Electronic structure
calculations confirm this ordering, the reverse of what has been proposed. NMR
results demonstrate a gapless Dirac semimetal state occurring at a Lifshitz
transition temperature, $T_c=85$ K in our crystals. We demonstrate that the
changes in NMR shift at $T_c$ also provide direct evidence of band inversion
when the topological phase transition occurs.
