Optomechanical cavities enables reversible coupling between light and mechanical motion. Furthermore, by harnessing the susceptibility of spin qubits to applied strain, optomechanical cavities constitutes a promising platform for the realisation of phonon mediated spin photon interfaces – a key element in a quantum network. Recently, a proof-of-principle of such a spin-photon interface was demonstrated using a diamond microdisk optomechanical cavity, where a single mechanical radial breathing mode, with a fixed mechanical frequency, was used to manipulate the spin-state of an ensemble of nitrogen-vacancy (NV) centers. However, this approach requires frequency matching between the mechanical frequency of the radial breathing mode and the qubit transition – a condition not always satisfied. In this work, we explore the potential of higher frequency mechanical modes, by performing multimode optomechanical spectroscopy of a diamond microdisks, accessible by the coherent coupling between optical cavity modes and mechanical resonances. In addition to the previously observed radial breathing mode, we have shown that the diamond microdisk exhibit a rich mechanical mode spectrum, with higher order mechanical resonances ranging in frequency from 1 GHz to 10 GHz vibrating in a different manner. Furthermore, the larger mechanical frequency offered by the higher order modes are less susceptible to decoherence due to their lower thermal phonon population. These mechanical modes exhibit Fano line shape, due to interference between the optical Kerr effect and the mechanical resonances. The Fano line shape allows the optomechanical coupling strength to be determined from the mechanical susceptibility, thus eliminating the need for employing traditionally frequency noise calibration technique that limits the measurement of the optomechanical coupling strength to the fundamental radial breathing mode. The observation of multiple mechanical modes which are modeled considering optical Kerr nonlinearity, offers a promising potential for improved spin-optomechanical experiments, in addition to expand applications using diamond microdisk to multimode optomechanics studies.