The flow of water in carbon nanochannels has defied understanding thus far, with accumulating experimental evidence for ultra-low friction, exceptionally high water flow rates, and curvature-dependent hydrodynamic slippage. These unique properties have raised considerable interest in carbon-based membranes for desalination, molecular sieving and energy harvesting. However, the mechanism of water-carbon friction remains unknown, with neither current theories, nor classical or ab initio molecular dynamics simulations providing satisfactory rationalisation for its singular behaviour. Here, we develop a quantum theory of the solid-liquid interface, which reveals a new contribution to friction, due to the coupling of charge fluctuations in the liquid to electronic excitations in the solid. We expect that this quantum friction, which is absent in Born-Oppenheimer molecular dynamics, is the dominant friction mechanism for water on carbon-based materials. As a key result, we demonstrate a dramatic difference in quantum friction between the water-graphene and water-graphite interface, due to the coupling of water Debye collective modes with a thermally excited plasmon specific to graphite. This suggests an explanation for the radius-dependent slippage of water in carbon nanotubes, in terms of the nanotubes' electronic excitations. Our findings open the way to quantum engineering of hydrodynamic flows through the confining wall electronic properties.