You can go with neutral/natural color. Or better pull any color from your rug.

Pull. the color from the rug. There are several shades of blue in it to pick from. Forest green definitely doesnt go with it, and from what I see of the rug, no shade of green goes with it.

I think blues and greens can work well together. I couldn’t find anything exactly like your room, just wanted to show the two together.

I look at rugs as floor art and don’t feel they have to perfectly match everything in certain cases.

I like this color because it has undertones of gray and brown. The rug has both of those tones and maybe this green mix will make a good compliment with the rug and tone down the strong color of the floor.