Are you in Canada now? Why don't you apply for OHIP now?
If you are not in Canada right now but are thinking of moving there before the baby is born, some provinces will cover you for health care the day you arrive if you are moving from another country. Some provinces will normally not but will make exceptions for childbirth. For OHIP see
http://www.health.gov.on.ca/english/public/program/ohip/ohipfaq_dt.html#Q2 to find out if you are exempt. If not, you can go to another province. Alberta will cover you from day one if you move from another country.