A direct numerical simulation (DNS) procedure is employed to study the thermal motion of a nanoparticle in an incompressible Newtonian stationary fluid medium with the generalized Langevin approach. We consider both the Markovian (white noise) and non-Markovian (Ornstein-Uhlenbeck noise and Mittag-Leffler noise) processes. Initial locations of the particle are at various distances from the bounding wall to delineate wall effects. At thermal equilibrium, the numerical results are validated by comparing the calculated translational and rotational temperatures of the particle with those obtained from the equipartition theorem. The nature of the hydrodynamic interactions is verified by comparing the velocity autocorrelation functions and mean square displacements with analytical results. Numerical predictions of wall interactions with the particle in terms of mean square displacements are compared with analytical results. In the non-Markovian Langevin approach, an appropriate choice of colored noise is required to satisfy the power-law decay in the velocity autocorrelation function at long times. The results obtained by using non-Markovian Mittag-Leffler noise simultaneously satisfy the equipartition theorem and the long-time behavior of the hydrodynamic correlations for a range of memory correlation times. The Ornstein-Uhlenbeck process does not provide the appropriate hydrodynamic correlations. Comparing our DNS results to the solution of an one-dimensional generalized Langevin equation, it is observed that where the thermostat adheres to the equipartition theorem, the characteristic memory time in the noise is consistent with the inherent time scale of the memory kernel. The performance of the thermostat with respect to equilibrium and dynamic properties for various noise schemes is discussed.