Drug delivery by intravascular use of targeted nanocarriers holds promise for personalized medicine, but clinical optimization of drug transport first requires an accurate description of carrier motion in the bloodstream and near endothelial cells. A synergistic computational approach is suitable to determine both the translational and rotational motions of a nanocarrier. Our formalism is primarily aimed at situations where both Brownian motion as well as the hydrodynamic interactions are important such as fluid motion in nanochannels and microcapillaries (0.2 – 20 micro m). In this study, a direct numerical simulation employing a finite element method has been implemented to simulate the Brownian motion of a nanoparticle in an incompressible Newtonian fluid medium. Two different approaches of assessing the motion of a nanoparticle in a fluid medium subject to thermal fluctuations are considered: (i) fluctuating hydrodynamics approach (Markovian fluctuating hydrodynamics of the fluid and a non-Markovian Ornstein-Uhlenbeck noise in the equations of motion of the particle), and, (ii) Langevin approach (Markovian thermostat or non-Markovian generalized colored noise for the equations of motion of the particle). The results obtained from both the approaches are validated by comparing the calculated temperature of the system, with that predicted by the equipartition theorem, and by comparing the velocity autocorrelation function with analytical results of Zwanzig and Bixon [Phys. Rev. A, 2, 2005-2012 (1970)]. The comparisons are excellent and found to be independent of mesh sizes and time step of integration. The simulations presented satisfy the generalized fluctuation-dissipation theorem for a range of memory correlation times in the generalized colored noise.