Kvasnička appeared in a solitary game on the blueline at the 2017 World Junior Championships. He later returned for the 2019 World Junior Championships in Vancouver, British Columbia. Kvasnička contributed with 3 points in 5 games, helping the Czech's advance to the knockout rounds, before a 3-1 elimination defeat to the United States on 2 January 2019. He was selected among the top 3 players for the Czech team in a 7th place finish.