In order to understand the spat, a bit of backstory is necessary. Hedo Turkoglu has been a loyal member of the Turkish national team. In fact, Turkoglu played for Turkey on 11 separate occasions from the years of 1999-2003. Enes Kanter, on the other hand, has not played for the Turkish national team since 2011. Additionally, Kanter has been publicly antagonistic towards the Turkish national team since 2013.
The latest development in this saga was when Enes Kanter got refused entry to the national team this Summer. Initially, Kanter cited politics as the reason why he couldn't play. The Turkish national government has been in conflict with the Gulen movement, which Kanter is a part of, since 2013. Eventually, Kanter tweeted a photo of Hedo Turkoglu with Turkey's president. The caption read something to the extent of, "this is why I was not invited".
Turkoglu took offense to what Kanter said, and responded with something to the extent of, "Enes' head does not work." Also, Turkoglu called for a permanent end to Kanter's career with the Turkish national team.
Heading into the 2015-16 NBA season, there is a potential for Turkoglu and Kanter to meet in a game. Obviously, Kanter is with the Thunder. Turkoglu is currently attempting to secure another contract, and the Kings are a distinct possibility. Were Turkoglu to sign with the Kings and get regular minutes, it's almost certain that he'd play against Kanter at some point. What a rivalry that would be!