When I had graves I didn't get the neck pain, not even when I developed a goiter, however, everyone is different.
I don't have Graves Disease, but it's my understanding that it wreaks havoc with the entire body, so I'd say "yes", it probably could make your neck hurt. Have you had an ultra sound to check for goiter or nodules?
I had both the uptake and ultrasound and all was negative. No nodules or goiter.