Luka Brajković began his basketball career with the Austrian team Dornbirn Lions, and in 2018 he moved across the Atlantic, where he spent four seasons playing for Davidson College. In his final season on American courts, he averaged 14.4 points, 7.1 rebounds, and 2.5 assists per game over 34 NCAA appearances. In 2022, he was also named the Atlantic 10 Conference Player of the Year.

After returning to Europe, he played for Spanish side Río Breogán in the 2022/23 season in the ACB League, followed by a stint in Greece with Kolossos during the 2023/24 season. In the 2024/25 campaign, he started the year with Italian club Tesi Pistoia, before moving to Turkish team Darüşşafaka in November. In the Italian domestic league, he appeared in eight games, averaging 8.9 points and 6.4 rebounds per game. In the Turkish league, he played 21 games, recording averages of 9.0 points and 5.6 rebounds per contest.

Brajković has been a part of the Austrian national team system for many years. He made his debut in FIBA competitions in 2015 at the Division B European Championship for the U16 team. As a member of the senior Austrian national team, he played in the pre-qualifiers for EuroBasket 2025 and the 2027 FIBA World Cup. In the latter, he averaged 15.3 points and 7.3 rebounds per game over four appearances.