Green tones from the carpet will complement the the large wood bookshelves. Deep red will complement the gray- green sofas and chairs. Make the color pop! Take your colors directly from the carpet.
With red you add excitement, passion, drama. Green tones work well against light wood and add rest and rejuvenation. What mood do you want to create?

The walls currently have green mossy tones, so the room is too monotone. I am afraid of a red. It is our family room where tons of kids hang out-they don't need any more energy. The room looks out to our back yard, so I like an outdoorsy feel.

Spanish Olive from Ben Moore would look good. It is a subtle and calming and would look good with the grey/green of your sofa.

Computer calibrations vary, but a gray green a bit darker than the sofa, but still light enough to not feel heavy would be calming. BM Paris Rain, Seattle Mist, Meditation, Wind Chime may work. Use the sofa pillow to find it's closest paint chip and go a shade or two darker from there. It will help eliminate having too yellow or bright of a green. Good luck :) !