Thiem defeated German qualifier Maximilian Marterer 7-5, 6-3 for a second-round meeting with Robin Haase or David Ferrer.
Monfils lost 3-6, 6-4, 6-3 to French compatriot Richard Gasquet, who saved 11 of the 13 break points he faced and won in just over two hours despite Monfils' 22 aces to his one.
Spain's Albert Ramos-Vinolas, the No. 8, was also beaten in the first round of the grass-court tournament, losing to Russian wild card Andrey Rublev 6-7 (1), 7-5 6-4.
Wild card Dustin Brown beat Canadian qualifier Vasek Pospisil 6-1, 6-7 (3), 7-6 (4), and Philipp Kohlschreiber, another German, defeated Portugal's Joao Sousa 3-6, 6-4, 6-4.
Eight-time champion Roger Federer opens his bid for his 16th grass-court title against Lu Yen-hsun of Taiwan on Tuesday.
Federer was surprised by German veteran Tommy Haas on his return from a two-month break last week in Stuttgart. The 35-year-old Swiss skipped the clay-court season to recuperate from a busy start to the year that saw him capture his 18th Grand Slam at the Australian Open. He hadn't played since winning the Miami Open in early April.