  We study non-equilibrium current fluctuations through a quantum dot, which
includes a ferromagnetic Hund's rule coupling $J$, in the low-energy Fermi
liquid regime using the renormalized perturbation theory. The resulting
cumulant for the current distribution in the particle-hole symmetric case,
shows that spin-triplet and spin-singlet pairs of quasiparticles are formed in
the current due to the Hund's rule coupling and these pairs enhance the current
fluctuations. In the fully screened higher-spin Kondo limit, the Fano factor
takes a value $F_b = (9M+6)/ (5M+4)$ determined by the orbital degeneracy $M$.
We also investigate crossover between the small and large $J$ limits in the
two-orbital case M=2, using the numerical renormalization group approach.
