No, it isn't.
The mixolydian mode in the key of C goes C, D, E, F, G, A, Bb. F is the 4, not the b7.
G mixolydian is the relative mixolydian scale of C major. That doesn't mean C major IS G mixolydian. This is the most common conceptual error about relative modes: that because they have the "same notes," they are the "same scale." They aren't. They are different scales.
"In the key of C" means exactly that: In the key of C – not in the key of G mixolydian. "C mixolydian" does NOT mean "the relative mixolydian scale of C major." It simply means C mixolydian.
F is the 7th interval of G mixolydian, or any other G-tonic scale that has a b7 in it. So just say that: "F is the 7 in the key of G mixolydian."