Yes, the player can have children or become pregnant. It can take a few days before you are informed of the pregnancy. Pregnancy without mods is simply a matter of having a penis cum inside a vagina repeatedly. Only contraceptives will prevent it. The note for the infertility trait explains in detail: https://strive4power.fandom.com/wiki/Traits#Infertile

If you have Aric's Expansion then pregnancy is more complex and there are ovulation cycles so pregnancy can only occur during certain intervals. A nurse can expose related information in the slave's statistics.