The nature of the melting transition of a bilayer comprising Hertzian spheres, confined between parallel walls, was investigated with the multicolour domain decomposition Monte Carlo simulation. It was confirmed that an ordered bilayer structure with a square lattice symmetry appeared parallel to the walls of confinement at a certain density and temperature. Bond-orientational order parameters of the system were studied numerically, and it has been suggested that this system obeys a two-stage melting transition with an intermediate phase called the tetratic phase. A snapshot of the spatial distribution of domains in which the bond-orientational order of the system exists shows a filamentary structure in the transition regime.