What you might consider before eliminating gluten is to ask your doctor to screen you for celiac disease. Both are autoimmune diseases commonly linked. Celiac testing (simple blood test) requires you to be consuming gluten.
If negative, give the GF diet a try. From what I have read (and am living) a Whole Foods diet (non-processed) is a good strategy for reducing inflammation and antibodies. Results vary.
I have Hashimoto's and celiac disease. My Mom has Graves. Please follow your doctor's advice about treatment (or get a second opinion). My mom sustained permanent eye damage from being undiagnosed for a long time.
Learn more about celiac disease from the University of Chicago.