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.
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