Can pulling out a green string from my nose cause any harm or worsen the results of my procedure?

Most likely these are sutures to close the mucosa inside the nose.  They are intended to dissolve and pulling them out as long as the incision line is healed, which it should be at 1 week, should present no problems. 

I would recommend contacting your surgeon to discuss the green string you pulled out of your nose after septorhinoplasty.  Only your surgeon will know what the green string was and what it's function may have been.  If the string was not attached to anything, it may not cause any harm, but again, only your surgeon will know for sure.