  We present the BRST formalism of a Weyl conformal gravity in Weyl geometry.
Choosing the extended de Donder gauge-fixing condition (or harmonic gauge
condition) for the general coordinate invariance and the new scalar
gauge-fixing for the Weyl invariance we find that there is a
Poincar${\rm{\acute{e}}}$-like $\IOSp(10|10)$ supersymmetry as in a Weyl
invariant scalar-tensor gravity in Riemann geometry. We also point out that
there is a gravitational conformal symmetry in quantum gravity although there
is a massive Weyl gauge field as a result of spontaneous symmetry breakdown of
Weyl gauge symmetry and account for how the gravitational conformal symmetry is
spontaneously broken to the Poincar\'e symmetry. The corresponding massless
Nambu-Goldstone bosons are the graviton and the dilaton. We also prove the
unitarity of the physical S-matrix on the basis of the BRST quartet mechanism.
