We assess the accuracy of various computational methods for obtaining infrared (IR) spectra of nanosized silicate dust grains directly from their atomistic structure and atomic motions. First, IR spectra for a selection of small nanosilicate clusters with a range of sizes and chemical compositions are obtained within the harmonic oscillator approximation employing density functional theory (DFT) based quantum chemical calculations. To check if anharmonic effects play a significant role in the IR spectra of these nanoclusters, we further obtain their IR spectra from finite temperature DFT-based ab initio molecular dynamics (AIMD). Finally, we also study the effect of temperature on the broadening of the obtained IR spectra peaks in larger nanosilicate grains with a range of crystallinities. In this case, less computationally costly classical molecular dynamics simulations are necessary due to the large number of atoms involved. Generally, we find that although DFT-based methods are more accurate, surprisingly good IR spectra can also be obtained from classical molecular dynamics calculations.