Quantum random number generators employ the inherent randomness of quantum mechanics to generate truly unpredictable random numbers, which are essential in cryptographic applications. While a great variety of quantum random number generators have been realised using photonics, few exploit the high-field confinement offered by plasmonics, which enables device footprints an order of magnitude smaller in size. Here we integrate an on-chip nanowire plasmonic waveguide into an optical time-of-arrival based quantum random number generation setup. Despite loss, we achieve a random number generation rate of $14.4\,\text{Mbits/s}$ using low light intensity, with the generated bits passing industry standard tests without post-processing. By increasing the light intensity, we were then able to increase the generation rate to $41.4\,\text{Mbits/s}$, with the resulting bits only requiring a shuffle to pass all tests. This is an order of magnitude increase in the generation rate and decrease in the device size compared to previous work. Our experiment demonstrates the successful integration of an on-chip nanoscale plasmonic component into a quantum random number generation setup. This may lead to new opportunities in compact and scalable quantum random number generation.