Quartz is an abundant mineral in Earth's crust whose mechanical behavior plays a significant role in the deformation of the continental lithosphere. However, the viscoplastic rheology of quartz is difficult to measure experimentally at low temperatures without high confining pressures due to the tendency of quartz (and other geologic materials) to fracture under these conditions. Instrumented nanoindentation experiments inhibit cracking even at ambient conditions, by imposing locally high mean stress, allowing for the measurement of the viscoplastic rheology of hard materials over a wide range of temperatures. Here we measure the indentation hardness of four synthetic quartz specimens and one natural quartz specimen with varying water contents over a temperature range of 23°C to 500°C. Yield stress, which is calculated from hardness but is model dependent, is fit to a constitutive flow law for low‐temperature plasticity to estimate the athermal Peierls stress of quartz. Below 500°C, the yield stresses presented here are lower than those obtained by extrapolating a flow law constrained by experiments at higher temperatures irrespective of the applied model. Indentation hardness and yield stress depend weakly on crystallographic orientation but show no dependence on water content.