Acoustic wave velocity measurement based on optical coherence tomography (OCT) is a promising approach to assess the mechanical properties of biological tissues and soft materials. While studies to date have demonstrated proof of concept of different ways to excite and detect mechanical waves, the quantitative performance of this modality as mechanical measurement has been underdeveloped. Here, we investigate the frequency dependent measurement of the wave propagation in viscoelastic tissues, using a piezoelectric point-contact probe driven with various waveforms. We found that a frequency range of 2-10 kHz is a good window for corneal elastography, in which the lowest-order flexural waves can be identified in post processing. We tested our system on tissue-simulating phantoms and ex vivo porcine eyes, and demonstrate reproducibility and inter-sample variability. Using the Kelvin-Voigt model of viscoelasticity, we extracted the shear-elastic modulus and viscosity of the cornea and their correlation with the corneal thickness, curvature, and eyeball mass. Our results show that our method can be a quantitative, useful tool for the mechanical analysis of the cornea.