The infrared bar-spectrum of a single ammonia molecule encapsulated in nano-cage C$_{60}$ fullerene molecule is modelled using the site inclusion model successfully applied to analyze spectra of CO$_2$ isotopologues isolated in rare gas matrix. Calculations show that NH$_3$ can rotate freely on a sphere of radius 0.184 Å around the site centre of the nano-cage and spin freely about its C$_3$ symmetry axis. In the static field inside the cage degenerate $\nu_3$ and $\nu_4$ vibrational modes are blue shifted and split. When dynamic coupling with translational motion is considered, the spectral signature of the $\nu_2$ mode is modified with a higher hindering barrier (2451 cm$^{-1}$), an effective reduced mass (6.569 g.mol$^{-1}$) and a longer tunneling time (55594 ps) for the fundamental level compared to gas-phase values (2047 cm$^{-1}$), (2.563 g.mol$^{-1}$) and (20.85 ps). As a result this mode is red shifted. Moreover, simulation shows that the changes in the bar-spectrum of the latter mode can be used to probe the temperature of the surrounding media in which fullerene is observed.