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Diffusion processes are useful tools for quantifying the dynamics of real world processes. By formulating a model in terms of a stochastic differential equation it is possible to gain insight into the dynamics of continuously evolving processes from discretely sampled trajectories using a compact set of equations. As such, diffusion models are extremely flexible and have found application in numerous fields of science. Perhaps one of the principle limitations in the use of diffusion models is the relatively sparse ecosystem of models with analytically tractable dynamics. As such, analysis has often focused on linear, time-homogeneous models. In this paper, we introduce the DiffusionRgqd package: A collection of tools for performing inference and analysis on scalar and bivariate time-inhomogeneous diffusion processes with quadratic drift and diffusion terms in R. The package focuses on likelihood based inference and model selection for discretely observed diffusion processes, and analysis based on quantities such as the transitional density and first passage time densities for scalar diffusion processes. We illustrate various features of the package by analysing a number of non-linear diffusion processes and performing inference on various diffusion models of a real-world dataset.