Can new birth control give me diarriah?

Usually no. Call your doctor who prescribed the medicine (any time you start a new medication and have symptoms, you should let them know). But the diarrhea is probably related to a viral infection, or reaction to some food.
No. Read the package insert to verify or look it up on the internet ( specific to the product you use). Contraceptives are not supposed to cause diarrhea as a common side effect ( to my knowledge).