Taking birth control for a month. could i be having diarrhea and vomiting from the birth control reacting to my period for the first time??

Possibly: Your body could be adjusting to the changes in hormones due to the birth control and this may cause some bowel symptoms. It may also be that you have a viral infection in the gut causing the symptoms and it's just coincidence that you are having a period at the same time. You'll have to see how your symptoms go over the next few months on the pill.