We introduce a new method for hardware nonuniform random number generation based on the transfer characteristics of graphene field-effect transistors (GFETs) which requires as few as two transistors and a resistor. We implement the method by fabricating multiple GFETs and experimentally validating that their transfer characteristics exhibit the nonlinearity on which our method depends. We use characterisation data in simulations of a proposed architecture for generating samples from dynamically selectable non-uniform probability distributions. The method we present has the potential for Gb/s sample rates, is reconfigurable for arbitrary target distributions, and has a wide range of possible applications. Using a combination of experimental measurements of GFETs under a range of biasing conditions and simulation of the GFET-based non-uniform random variate generator, we demonstrate a speedup of Monte Carlo integration by up to 2×. This speedup assumes the analog-to-digital converters reading the outputs from the circuit can produce samples in the same amount of time that it takes to perform memory accesses.