  The quantum version of the Bernstein-Gelfand-Gelfand resolution is used to
construct a Dolbeault-Dirac operator on the anti-holomorphic forms of the
Heckenberger-Kolb calculus for the $B_2$-irreducible quantum flag manifold. The
spectrum and the multiplicities of the eigenvalues of the Dolbeault-Dirac
operator are computed. It is shown that this construction yields an
equivariant, even, $0^+$-summable spectral triple.
