Ferrer is playing his first tournament since losing to Rafael Nadal in French Open semifinals earlier this month. The Spaniard cruised past Mayer in an hour and next plays Igor Sijsling, the only Dutch player left in the grass-court prep event for Wimbledon.
Ferrer is seeded seventh for Wimbledon, which starts on Monday.
On the women's side, Francesca Schiavone of Italy saved four match points during a 6-2, 2-6, 7-6 (8) victory over Irina-Camelia Begu of Romania.
The 2010 French Open champion made it tough on herself, serving 14 double faults and falling behind 6-2 in the tiebreaker before fighting back to win.
In other men's matches, Tatsuma Ito of Japan needed less than an hour to beat third-seeded Austrian Jurgen Melzer 6-2, 6-2, and Frenchman Benoit Paire beat eighth-seeded Lukasz Kubot of Poland 6-3, 0-6, 6-4.