Improving the quality factor of the mechanical oscillations of micron scale beams in a viscous fluid, such as water, is an open challenge of direct relevance to the development of future technologies. We study the stochastic dynamics of doubly-clamped micron scale beams in a viscous fluid driven by Brownian motion. We use a thermodynamic approach to compute the equilibrium fluctuations in beam displacement that requires only deterministic calculations. From calculations of the autocorrelations and noise spectra we quantify the beam dynamics by the quality factor and resonant frequency of the fundamental flexural mode over a range of experimentally accessible geometries. We carefully study the effects of the grid resolution, domain size, linear response, and time-step for the numerical simulations. We consider beams with uniform rectangular cross-section and explore the increased quality factor and resonant frequency as a baseline geometry is varied by increasing the width, increasing the thickness, and decreasing the length. The quality factor is nearly doubled by tripling either the width or the height of the beam. Much larger improvements are found by decreasing the beam length, however this is limited by the appearance of additional modes of dissipation. Overall, the stochastic dynamics of the wider and thicker beams are well predicted by a two-dimensional approximate theory beyond what may be expected based upon the underlying assumptions, whereas the shorter beams require a more detailed analysis.