Semiconductor InSb nanowires are expected to provide an excellent material platform for the study of Majorana fermions in solid state systems. Here, we report on the realization of a Nb-InSb nanowire-Nb hybrid quantum device and the observation of a zero-bias conductance peak structure in the device. An InSb nanowire quantum dot is formed in the device between the two Nb contacts. Due to the proximity effect, the InSb nanowire segments covered by the superconductor Nb contacts turn to superconductors with a superconducting energy gap Δ(InSb) ∼ 0.25 meV. A tunable critical supercurrent is observed in the device in high back gate voltage regions in which the Fermi level in the InSb nanowire is located above the tunneling barriers of the quantum dot and the device is open to conduction. When a perpendicular magnetic field is applied to the devices, the critical supercurrent is seen to decrease as the magnetic field increases. However, at sufficiently low back gate voltages, the device shows the quasi-particle Coulomb blockade characteristics and the supercurrent is strongly suppressed even at zero magnetic field. This transport characteristic changes when a perpendicular magnetic field stronger than a critical value, at which the Zeeman energy in the InSb nanowire is E(z) ∼ Δ(InSb), is applied to the device. In this case, the transport measurements show a conductance peak at the zero bias voltage and the entire InSb nanowire in the device behaves as in a topological superconductor phase. We also show that this zero-bias conductance peak structure can persist over a large range of applied magnetic fields and could be interpreted as a transport signature of Majorana fermions in the InSb nanowire.