On 3 June 2007, Božović made his international debut for Montenegro in a 0–1 loss to Colombia at the Kirin Cup. He then served as a backup goalkeeper for Vukašin Poleksić during the team's unsuccessful qualification campaign for the 2010 FIFA World Cup. Subsequently, Božović played the full 90 minutes in all of his team's UEFA Euro 2012 qualifying games, including a play-offs tie against the Czech Republic. However, Montenegro failed to reach the final tournament.