We have investigated the elastic response of a transverse Ising magnet CoNb2O6 by means of ultrasound velocity measurement. A huge elastic anomaly in the C66 mode is observed near a quantum critical point when sweeping a magnetic field perpendicular to the Ising axis. This anomaly appears to become critical only for the Faraday configuration (field parallel to the sound propagation direction) but is much less pronounced for the Voigt geometry (field perpendicular to the sound propagation direction). We propose that the relativistic spin-orbit interaction plays a crucial role in the quantum critical regime resulting in the elastic anomaly, which is enhanced by quantum fluctuations.