The following is what I have found, which shows promise on your hypothesis.
Yanmaz, at al. (2012) found
an increased prevalence rate of [Fibromyalgia Syndrome] in patients with [diabetes mellitus] type 2 and [rheumatoid arthritis]. There was no correlation between the prevalence rate of [Fibromyalgia Syndrome] and good [diabetes mellitus] disease control.
A pilot study by Mengshoel, et al. (1995) followed a 10-week multi-disciplinary approach which involved exercise and cognitive approaches.
For the cognitive approach, they looked at patient knowledge of fibromyalgia and instructed on diet.
According to the food records, the dietician informed about good food habits and suggested dietary improvements such as: intake of four regular meals a day, increased intake of fruits and vegetables, increased intake of fluid and reduced intake of sweets and sugar.
All patients (n=16)
showed improvements for all variables after ten weeks.
They also said that controlled studies are needed in the future.
Mengshoel, A. M., Forseth, K. Ø., Haugen, M., Walle-Hansen, R., & Førre, Ø. (1995). Multidisciplinary approach to fibromyalgia A pilot study. Clinical rheumatology, 14(2), 165-170. doi: 10.1007/BF02214937
Yanmaz, M. N., Mert, M., & Korkmaz, M. (2012). The prevalence of fibromyalgia syndrome in a group of patients with diabetes mellitus. Rheumatology international, 32(4), 871-874. doi: 10.1007/s00296-010-1618-8