Theory predicts that a large spontaneous electric polarization and concomitant inversion symmetry breaking in GeSe monolayers result in a strong shift current in response to their excitation in the visible range. Shift current is a coherent displacement of electron density on the order of a lattice constant upon above-bandgap photoexcitation. A second-order nonlinear effect, it is forbidden by the inversion symmetry in the bulk GeSe crystals. Here, we use terahertz (THz) emission spectroscopy to demonstrate that ultrafast photoexcitation with wavelengths straddling both edges of the visible spectrum, 400 and 800 nm, launches a shift current in the surface layer of a bulk GeSe crystal, where the inversion symmetry is broken. The direction of the surface shift current determined from the observed polarity of the emitted THz pulses depends only on the orientation of the sample and not on the linear polarization direction of the excitation. Strong absorption by the low-frequency infrared-active phonons in the bulk of GeSe limits the bandwidth and the amplitude of the emitted THz pulses. We predict that reducing GeSe thickness to a monolayer or a few layers will result in a highly efficient broadband THz emission. Experimental demonstration of THz emission by the surface shift current in bulk GeSe crystals puts this 2D material forward as a candidate for next-generation shift current photovoltaics, nonlinear photonic devices, and THz sources.