I'm thinking gray, tan, or in this neutral family. You can steer towards muted versions of a greenish beige, or a bluish gray to bring out the tones of the turquoise.

Here's a turquoise ideas book that has some rooms that contrast beautifully with neutral grays or tans.

https://www.houzz.com/magazine/color-12-ways-to-love-turquoise-stsetivw-vs~94587

Thanks so much!!

You're welcome! :)

Or you could spend a few hundred hours looking at this blog:
http://www.houseofturquoise.com/