I am 20 years old, is a menstrual cycle during pregnancy is hereditary? My mother swears she had four periods while pregnant with me.

Your mom is wrong. It is absolutely impossible to have a "period" when pregnant. It is possible to have a bit of "spotting" for a variety of reasons but true bleeding (as intense as a period) is almost always a sign of serious problems. See http://www.Whattoexpect.Com/pregnancy/symptoms-and-solutions/vaginal-spotting.Aspx.
No. Your mothers episodes of bleeding were not from the same reasons a non-pregnant women has her menstrual cycle. Most often the "period" like bleeding in pregnancy is coming from the placenta and can be a cause of danger. Fortunately it appears all went well.