After Safiullin claimed the first set Muller, who has only two career wins on grass, took a two-game lead in the second but Safiullin recovered to clinch it and will next face Serbia's Hamad Medjedovic.
Muller's compatriot Moutet, fresh off his stunning victory over Taylor Fritz at Queen's last week, struggled with frustration in the first set against Martinez, smashing rackets and voicing his anger before losing the opener.
The second set saw a more composed Moutet, who capitalised on Martinez's injury concerns to level the match before sealing a berth in the next round with a bagel in the decider.
Frenchman Arthur Rinderknech beat Bosnian veteran Damir Dzumhur 6-4 4-6 7-5 to next meet Canadian second seed Felix Auger Aliassime while sixth seed Gabriel Diallo defeated Spain's Jaume Munar 7-5 6-3 to book a meeting with Serbian Laslo Djere.
Tuesday's play opens with the all-Australian clash between 32-year-old qualifier Bernie Tomic, playing his first tour-level event for the first time since he qualified for the 2021 Australian Open, and Rinky Hijikata.
At Eastbourne Briton Dan Evans, 170 in the world, battled back from a set down to defeat Miomir Kecmanovic in Monday's first round.
The 35-year-old, who has received a wild card for Wimbledon, triumphed 3-6 6-4 6-4 against the Serbian world No. 49.
He will face American second seed Tommy Paul - the 2023 runner-up - in round two.
