Yes. Menstrual cycles are so different is every woman, but you also need to make sure you are not becoming anemic or developing a nutritional or mineral deficiency when you menstruate. Vitamin or folate (folic acid) or iron deficiency can all cause gingival (gum) pain. See your ob/gyn and possibly consider further work up to ensure no deficiency exists.
Hormones. Hormones affect gums, that is the most likely gums. The healthier your gums are the less likely that is to happen. Do you have regular professional care. If you do not, please seek it out.
Sore Gums. Sore gums mean irritated or inflamed gums. This may happen briefly during the mid-cycle hormonal surge, but is uncommonly the cause of sore gums. Go see your dentist, if this symptom doesn't clear or recurs to make sure of the cause.