The motion of a Brownian particle in the presence of a deformable interface is studied by considering the random distortions of interface shape due to spontaneous thermal impulses from the surrounding fluid. The fluctuation-dissipation theorem is derived for the spontaneous fluctuations of interface shape using the method of normal modes in conjunction with a Gangevin type equation of motion for a Brownian particle, in which the fluctuating force arises from the continuum motions induced near the particle by the fluctuation of interface shape. The analysis results in the prediction of autocorrelation functions for the location of the dividing surface, for the random force acting on the particle, and the particle velocity. The particle velocity correlation, in turn, yields the effective diffusion coefficient due to random fluctuations of the interface shape.