  We develop a new framework of the self-consistent deformed proton-neutron
quasiparticle-random-phase approximation (pnQRPA), formulated in the
Hartree-Fock-Bogoliubov (HFB) single-quasiparticle basis. The same Skyrme force
is used in both the HFB and pnQRPA calculations except in the proton-neutron
particle-particle channel, where an S=1 contact force is employed. Numerical
application is performed for Gamow-Teller (GT) strength distributions and
$\beta$-decay rates in the deformed neutron-rich Zr isotopes located around the
path of the rapid-neutron-capture process nucleosynthesis. It is found that the
GT strength distributions are fragmented due to deformation. Furthermore we
find that the momentum-dependent terms in the particle-hole residual
interaction leads to a stronger collectivity of the GT giant resonance. The T=0
pairing enhances the low-lying strengths cooperatively with the T=1 pairing
correlation, which shortens the $\beta$-decay half lives by at most an order of
magnitude. The new calculation scheme reproduces well the observed isotopic
dependence of the $\beta$-decay half lives of deformed $^{100-110}$Zr isotopes.
