The hydrogenated antimonene (SbH) monolayer was predicted to be a 2D topological insulator with a large nontrivial band gap. It was also predicted to have high dynamic stability and could be applied to nanodevices at room temperature. Here, we studied the electronic structure and the quantum transport in the zigzag SbH nanoribbons (Z-SbH-NRs) based on the combination of density functional theory and the nonequilibrium Green's function method. Our calculations suggest that there is an interesting odd−even oscillating band gap in the Z-SbH-NRs. The oscillation may be related to the structure of the nanoribbons and the hybridization of the adjacent orbitals of Sb atoms. Based on the odd−even oscillating band gap, we constructed concave Z-SbH-NRs to further study the transport of topological edge states. The transport can be completely blocked in the concave Z-SbH-NRs. However, single Sn-doping can lead to perfect resonance tunneling in the Sn-doped concave Z-SbH-NRs. Our calculations may be helpful for applications of topological nanoribbons in nanodevices and spintronics.