Magnetic field emerging from the geometric curvature of quantum structure of electrons in a crystal bends electronic trajectory causing anomalous linear and nonlinear electrical Hall effects that have been observed in low symmetry crystals with narrow electronic band gap. We present first-principles theoretical analysis to show that dynamical lowering of crystal symmetry by lattice vibrations results in oscillations in the quantum geometry of electrons which have observable nonlinear Hall signatures. Using these, we introduce a vibrational spectroscopy based on Geometry of Quantum Electronic Structure (GQuES) making specific predictions for the transport and radiative GQuES spectra of 2D materials. As the crystal symmetry is commonly lowered by dynamical fields, GQuES is applicable to a wide range of materials and excitations spanning sub-GHz, THz and infrared frequencies.