Can a menstrual cycle start in one month and enter another month?

Yes : Menstrual cycles work by days, not months. The "cycle" is preparation for implantation, ovulation, removal of unused uterine lining that was ready for implantation. The timing is by days 11-14 days after the previous menses ovulation occurs. If no implantation happens the period starts about 12-14 days later. So your period could start on the last day of a month and continue onto the next month.