  We present a new formula to calculate matrix elements of a general unitary
operator with respect to Hartree-Fock-Bogoliubov states allowing multiple
quasi-particle excitations. The Balian-Br\'ezin decomposition of the unitary
operator (Il Nuovo Cimento B 64, 37 (1969)) is employed in the derivation. We
found that this decomposition is extremely suitable for an application of
Fermion coherent state and Grassmann integrals in the quasi-particle basis. The
resultant formula is compactly expressed in terms of the Pfaffian, and shows
the similar bipartite structure to the formula that we have previously derived
in the bare-particles basis (Phys. Lett. B 707, 305 (2012)).
