The phrase "quarter finalist" refers to one of the eight participants who have reached the quarterfinals of a competition. It means they are among the best eight teams or individuals in the tournament. Full definition
The complete list of quarter finalists can be found HERE. (goodereader.com)
In 2001 he was a semi finalist and quarter finalist in 2000 in the Quiksilver Phuket Contest. (surfcamp-thailand.com)
Leicester are the current league champions and champions league quarter finalists, they had a whole week, well rested to prepare for this game. (justarsenal.com)