Short answer; sometimes, yes.
As you probably already know, celiac disease is an autoimmune disorder.
Autoimmune disorders are those in which the body mistakenly attacks its own tissues. People who have one autoimmune disorder are at increased risk of developing other autoimmune disorders, perhaps because there is some underlying problem typically involved in autoimmune disorders.
Arthritis is another autoimmune disorder, and there are reports occasionally that some people affected by arthritis see dramatic improvement when they adopt a gluten free diet.
In my own case I certainly did have joint pain before I went GF over ten years ago. In particular I would get terrible pain in my ankles if I stepped off a curb too quickly or jumped a short distance that would never hurt most people. Any jarring of the bones in my legs would cause me to ache for hours or days. I think my bones and joints were becoming soft as a result of malnutrition.
Over time my joint pain diminished and now I seldom experience joint pain, and if I do it is because of some minor injury caused by my remarkable clumsiness and lack of co-ordination. I can run in place while exercising with no soreness at all now. And my Wii Fit will be the first to tell you that I'm not undernourished these days, I must have gained 30 pounds since I went GF.
You may find that the GF diet relieves your joint pain, many people do. I hope so

Steve