Murray, having claimed his first title on home soil since Wimbledon in 2016 with his win at Surbiton last week, secured an early break to take control of the first set against Grenier.
The Frenchman came out firing in the second but the Scot rallied with a hold to love to draw level at 5-5 before following up with a break and then serving out the match to secure a straight-set win.
He is ranked 44th in the world and trying to secure a top-32 seeding for Wimbledon, which begins on 3 July.
“It’s been a lot of matches for me,” said Murray. “I’m an old boy now so it takes me a bit longer to recover but I’m feeling good.”
Murray joins fellow Briton George Loffhagen in Friday’s quarter-finals and said it was exciting to see so many home players having success at the tournament.
“It’s brilliant, especially for the women, they’ve done well this week. A lot of them have had injury problems but they’ve all started the grass-court season well. Let’s keep it going into the weekend.”
Anther Briton, Ryan Peniston, ranked 201 in the world, lost 6-4 6-2 to Switzerland’s number eight seed Dominic Stricker – who Murray will now face not before 16:00 BST on Friday.