We present Quokka, a new subcycling-in-time, block-structured adaptive mesh refinement (AMR) radiation hydrodynamics code optimised for graphics processing units (GPUs). Quokka solves the equations of hydrodynamics with the piecewise parabolic method (PPM) in a method-of-lines formulation, and handles radiative transfer via the variable Eddington tensor (VET) radiation moment equations with a local closure. We use the AMReX library to handle the adaptive mesh management. In order to maximise GPU performance, we combine explicit-in-time evolution of the radiation moment equations with the reduced speed-of-light approximation. We show results for a wide range of test problems for hydrodynamics, radiation, and coupled radiation hydrodynamics. On uniform grids in 3D on a single GPU, our code achieves > 250 million hydrodynamic updates per second and almost 40 million radiation hydrodynamic updates per second. For radiation hydrodynamics problems on uniform grids in 3D, our code scales from 4 GPUs to 256 GPUs with an efficiency of 76 per cent. The code is publicly released under an open-source license on GitHub.