I awarded points based on slam finishs: 1 for quarters, 2 for semis, 3 for final, and 5 for the champion. This is men's and women's combined.
Incredibly, since I collecting this data in 2009, each year except one (2013) 22 countries have made an appearance in a slam quarterfinal.
Points
Switzerland (15 = 7 qtrs, 4 semis, 2 finals, 1 champ): Federer 7, Wawrinka 7, Bencic
Serbia (12 = 5 qtrs, 3 semis, 2 finals, 1 champ): Djokovic 11, Ivanovic
Spain (12 = 6 qtrs, 2 semis, 2 finals, 1 champ): Nadal 8, Ferrer 2, Muguruza, Suarez Navarro
Czech Republic (12 = 6 qtrs, 3 semis, 1 final, 1 champ): Kvitova 5, Berdych, Safarova 2, Zahlavova Strycova
Canada (10 = 5 qtrs, 4 semis, 1 final): Bouchard 7, Raonic 3
Russia (9 = 4 qtrs, 2 semis, 1 final, 1 champ): Sharapova 5, Makarova 3, Kuznetsova
China (7 = 2 qtrs, 2 semis, 1 final): Li 5, Peng 2
Croatia (6 = 2 qtrs, 1 semi, 1 final, 1 champ): Cilic 6
Romania (6 = 3 qtrs, 2 semis, 1 final): Halep 6
United States (5 = 1 qtr, 1 semi, 1 final, 1 champ): S.Williams 5
United Kingdom (5 = 4 qtrs, 1 semi): Murray 5
Germany (4 = 3 qtrs, 1 semi): Petkovic 2, Kerber, Lisicki
Italy (4 = 4 qtrs): Errani 2, Pennetta 2
Denmark (3 = 1 qtr, 1 semi, 1 final): Wozniacki 3
Japan (3 = 1 qtr, 1 semi, 1 final): Nishikori 3
Slovakia (3 = 1 qtr, 1 semi, 1 final): Cibulkova 3
Bulgaria (3 = 2 qtrs, 1 semi): Dimitrov 3
Latvia (2 = 1 qtr, 1 semi): Gulbis 2
Poland (2 = 1 qtr, 1 semi): Radwanska 2
Belarus (2 = 2 qtrs): Azarenka 2
France (2 = 2 qtrs): Monfils 2
Australia (1 = 1 qtr): Kyrgios
Number of Different Players to reach a quarterfinal
Czech Republic: 4
Spain: 4
Germany: 3
Russia: 3
Switzerland: 3
Canada: 2
China: 2
Italy: 2
Serbia: 2