I don’t think the rings would be close enough for easily detectable effects, but in terms of Jupiter in general, the answer is yes, and Juno is on the job.

The Juno spacecraft's suite of science instruments will primarily characterize and explore the three-dimensional structure of Jupiter's polar magnetosphere, auroras and mass composition.[4] As Juno is a Polar Orbit mission, it will be possible to measure the orbital frame-dragging, known also as Lense–Thirring precession, caused by the angular momentum of Jupiter.[5]

https://en.m.wikipedia.org/wiki/Frame-dragging#Experimental_...

The Schwarzschild radius of Saturn is 3 *(Msat/MSun) km ~ 0.85 m, so it's much smaller than Saturn Radius ~ 6x10^7 m. As such, I believe any GR effect is probably absolutely negligible at the location of Saturn's rings.

Well even in earths orbit GR is not negligible, say even for the purposes of GPS