Can a girl get pregnant if she is on birth control and got her period the same day she always gets it every month?

Much depends on the type of birth control.Not all birth control is effective 100% of the time.Medications can interfere with the pill,condoms can break.
If she is on an effective birth control and had a regular period,I would say chances are good that there is no pregnancy.
Your doctor is your best resource for medical advice.