Early career in Serbia Brkić, playing in the goalkeeper position, started his career as a professional began at the club
Indeks Novi Sad and
Vojvodina, before playing professionally at Vojvodina. Brkić loaned him to
Proleter Novi Sad in 2005, where he stayed for the club about two years. At Proleter Novi Sad, Brkić made thirty-one appearances. After his return from loan, Brkić made his debut for the club, in a 1–0 win over
Napredak Kruševac on 2 September 2007. Since then, Brkić soon became a first-choice goalkeeper and soon became the club's captain. At the club, Brkić made a total of 117 appearances.
Siena In the summer of 2011, Brkić left Serbia to join Italian side
Udinese. Because Udinese ran out of non-
EU registration quota for international signing (which Udinese had to buy one from other club) as well as the performance of Slovenian internationals
Samir Handanović, Brkić was signed by
Serie A club
Siena in definitive deal). On 11 September 2011, Brkić made his debut in a 0–0 draw against
Catania, keeping his first clean sheet on his debut in Serie A. In October, following impressive performances with Siena, Brkić was linked with a move to English side
Tottenham Hotspur. In response of a transfer speculation, however, Brkić told Serbian daily
Sport that he is aware of the transfer rumour, but insists he is focusing at Siena. Brkić also says he is no rush of leaving the club and insists the move is not affecting his form. After a 2–0 loss against
Milan on 17 December 2011 in which Brkić get a yellow card resulting a penalty, Brkić soon had an operation that kept him out for many months after fracturing his left fibula. A year on, Brkić spoke about his injury and find it difficult to come back to recover. In January 2012, Brkić was transferred back to Udinese for free Having established himself in the first team, Brkić made his debut for the club in the
UEFA Champions League qualification round against Portuguese side
Braga. Unfortunately, Udinese was eliminated in the Champions League and demoted to the
UEFA Europa League after losing in the penalty shoot-out in the second leg, just three days before he made his league debut in a 2–1 loss against
Fiorentina. However, the next game was even went worse for Brkić when he gave away the penalty in early minutes and received a red card in a 4–1 loss against regaining champion
Juventus. After the match, former referee Daniele Tombolini says Brkić's sending-off was a mistake because
Sebastian Giovinco, the opposition player in the match, did not have possession of the ball, though he maintained his views that the foul was a penalty. Brkić then miss one match against Siena and made his return when Udinese won 2–1 against Milan the next game. On 4 October 2012, Brkić was in goal when Udinese beat
Liverpool 3–2, making good save from
Luis Suárez twice and a header from
Jonjo Shelvey. Unfortunately, Udinese would then be eliminated from the Europa League, following a strings of losses. In mid-January, Brkić soon suffered an injury after fracturing the second finger of his right hand and went to surgery that kept him out at least a month. A month later, Brkić made his return, playing in friendly match against Gemonese on 26 February 2013. After the match, Brkić says he was prepared to make a comeback. Brkić made his return on 3 March 2013 against
Pescara, where the club won 1–0. After the match, Brkić returned as a first-choice goalkeeper for the rest of the
2012–13 season.
2013–14 In the
2013–14 season, Brkić was ruled out for the start of the season after injuring his shoulder while exercising in the gym, which ruled him out for four months. During his absence, he was replaced by
Simone Scuffet in goal. After three months out of action, Brkić made his return in a 2–1 win over
Sassuolo. Brkić's performance with Udinese were praised in the media, who noted that he "saved at least three goal-bound efforts while blocking several strong shots and commanded his area giving sustenance and substance to his whole defence". This earned him Serie A Team of Week 10.
Juventus invited schoolchildren to watch their match against Udinese in December in place of banned Ultras, but they shouted abuse at Brkić after each goal kick, and Juventus were fined €5,000.
PAOK On 7 July 2016, Brkić signed a three-year deal with
Super League Greece club
PAOK. Brkić, who had already agreed terms with PAOK's president/technical director
Ľuboš Micheľ, passed his medical at the
Thessaloniki-based club and then signed his new contract, leaving Italy after five consecutive seasons. Brkić, along with the experienced Greek internationals
Panagiotis Glykos and
Markos Vellidis, is expected to be part of a very strong
triplete in goalkeeper's position for PAOK ahead of
2016–17 season. ==International career==