Insulating magnets can display novel signatures of quantum fluctuations as similar to the case of metallic magnets. However, their weak spin-lattice coupling has made such observations challenging. Here we find that antiferromagnetic (AF) quantum fluctuations manifest in the dielectric properties of multiferroic Ba2CoGe2O7, where a ferroelectric polarization develops concomitant to an AF ordering. Upon application of a magnetic field (H), dielectric constant shows a characteristic power-law dependence near absolute zero temperature and close to the critical field Hc=37.1 T due to enhanced AF quantum fluctuations. When H>Hc, the dielectric constant shows the temperature-dependent anomalies that reflect a crossover from a field-tuned quantum critical to a gapped spin-polarized state. We uncover theoretically that a linear relation between AF susceptibility and dielectric constant stems from the generic magnetoelectric coupling and directly explains the experimental findings, opening a new pathway for studying quantum criticality in condensed matter.