We compute the charm-quark current-current correlation function within the Langevin framework and extract the heavy-quark diffusion coefficient using the Green-Kubo formula. The formalism is further extended to evaluate the correlation function using a generalized Langevin equation that incorporates memory effects via an exponentially decaying memory kernel. We find that while memory effects qualitatively modify the transient structure of the current correlations, the value of the transport coefficient remains unchanged. In addition, we investigate heavy-quark thermalization in the presence of memory and compare the nonequilibrium solution of a generalized Fokker-Planck equation with the one obtained from the generalized Langevin equation in the non-relativistic limit. We also consider the relativistic case with correlated noise and observe that memory effects can give rise to a damped, oscillatory equilibration of the heavy quark momentum distribution in a non-Markovian bath.