Suarez Navarro is one of only two seeds remaining in the outdoor clay competition. Sixth-seeded Maria Kirilenko of Russia, the defending champion, reached the last eight on Wednesday.
No. 7 Sara Errani of Italy was eliminated by the Czech Republic's Lucie Safarova 6-3, 6-3 on a rain-affected day.
Italy's Francesca Schiavone also advanced by beating Nicole Vaidisova of the Czech Republic 6-2, 6-3.
On Friday, Suarez Navarro will meet Germany's Tatjana Malek, who advanced after France's Stephanie Cohen-Aloro retired when trailing 5-0 in the first set of their match.
In the other quarterfinals, Safarova meets Schiavone, Kirilenko faces Maria Jose Martinez Sanchez of Spain, and Anastasiya Yakimova of Belarus plays Roberta Vinci of Italy.
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