The shear behavior of monolayer water films confined in a slit-like pore between hydrophilic walls is simulated in the quasistatic regime using the grand canonical Monte Carlo technique. Each wall is represented as a hexagonal lattice of force sites that interact with water through an orientation-dependent hydrogen-bonding potential. When the walls are in registry, the water oxygen atoms form either a crystal- or fluid-like structure, depending on the period of the wall's lattice. In both cases, however, the monolayer structure is orientationally disordered. Both the crystal-and fluid-like monolayers prove to be capable of experiencing well-defined stick-slip transitions, with the largest yield stress occurring in the crystal-like case. Beyond the yield point, the crystal-like monolayers "melt", but their structure and molecular motion differ in many respects from those characteristic of normal fluids.