  We investigate ground-state properties of a $t_{\rm 2g}$-orbital Hubbard
model on a zigzag chain relevant for CaV$_{2}$O$_{4}$, by exploiting numerical
techniques such as Lanczos diagonalization and density-matrix renormalization
group. Assuming a V$^{3+}$ ion, a local spin $S$=$1$ state is formed by two
electrons in the $t_{\rm 2g}$ orbitals. That is, the system is a Haldane system
with active $t_{\rm 2g}$-orbital degrees of freedom. We observe orbital-state
transitions, yielding a distinct spin system under the orbital-ordered
background. We also discuss the orbital structure induced by open edges,
originating in the spatial anisotropy of the $t_{\rm 2g}$ orbitals.
