We present the manifestly covariant canonical operator formalism of a Weyl invariant (or equivalently, a locally scale invariant) gravity whose classical action consists of the well-known conformal gravity and Weyl invariant scalar-tensor gravity, on the basis of the Becchi-Rouet-Stora-Tyupin (BRST) formalism. It is shown that there exists a Poincar${\rm{\acute{e}}}$-like $\mathit{IOSp}(8|8)$ global symmetry as in Einstein's general relativity, which should be contrasted to the case of only the Weyl invariant scalar-tensor gravity where we have a more extended Poincar${\rm{\acute{e}}}$-like $\mathit{IOSp}(10|10)$ global symmetry. This reduction of the global symmetry is attributed to the presence of the St\"{u}ckelberg symmetry.
