Murray sent shockwaves through the tennis world when he announced on Sunday that Frenchwoman Mauresmo would replace Ivan Lendl, who quit in March, as his coach for at least the grass-court season. (Agencies)
The 27-year-old's decision to ignore several high-profile male coaches to hire Mauresmo, a former world number one and Wimbledon champion, has been the talk of the sport for the last three days.
Mauresmo, the current France Fed Cup captain, has tried her hand at coaching on several occasions since retiring in 2009.
She had a brief spell working with male player Michael Llodra in 2010, but in the testosterone fuelled world of men's tennis few leading players have female coaches and the decision is seen in some quarters as a major gamble.
In the circumstances, a routine second round victory at this Wimbledon warm-up was exactly the kind of first date the pair needed as Murray, bidding to win Queen's for a record-equalling fourth time, proved far too strong for world number 92 Mathieu.
Since losing to Roger Federer in the 2012 Wimbledon final, Murray has been unstoppable on grass and his winning run, which brought him Olympic gold, a first Wimbledon crown and a third Queen's title, has now extended to 19 matches.
Murray's next opponent in the third round will be Czech 15th seed Radek Stepanek, who defeated Australian Bernard Tomic 7-6 (7/4), 7-6 (7/5).
Asked what Mauresmo would make of his performance, Murray told the BBC: "I don't know, I'll go and speak to her and see what she thinks.
"I served pretty well. It was a good start to the grass-court season. I have great memories here and always enjoy coming back."
Murray emerged to warm applause sporting a neat new haircut, while photographers focused their lens on Mauresmo, wearing a black T-shirt, as she took her place next to assistant coach Dani Vallverdu at one side of the players' box, with the Scot's girlfriend Kim Sears at the other end.
Mauresmo, who will work alongside Vallverdu, fitness trainers Matt Little and Jez Green and physio Mark Bender, had overseen Murray's practice session for the first time earlier in the day and the world number five looked at home back on grass following his humbling French Open semi-final thrashing against Rafael Nadal last week.
Murray sent shockwaves through the tennis world when he announced on Sunday that Frenchwoman Mauresmo would replace Ivan Lendl, who quit in March, as his coach for at least the grass-court season.