Born in Titovo Užice, Brković came through the youth system at Partizan, but never made it into the senior squad. He was instead loaned to affiliated club Teleoptik in order to gain experience. Before moving abroad, Brković made his Serbian SuperLiga debuts with Hajduk Kula and Smederevo.
In the summer of 2012, Brković signed for Israeli club Hapoel Haifa. He spent one year there, winning the Toto Cup, before moving to Hungarian side Debrecen. In his debut season at the club, Brković helped them win the league title.
After a brief spell at Latvian club Riga, Brković returned to Hungary and joined Diósgyőr in the 2018 winter transfer window. He spent there two and a half years, becoming leader of the team and captain. After expiring of the contract he left the Hangary.
