Kids can be found in Stardew Valley from the start, but having kids of your own is also possible. In order to have kids you first need to have a wife or a husband. They also need to be happy, meaning they have at least 12 hearts. To build a nursery you need to visit Robin's Shop and to get the third and final upgrade (that adds the nursery), you will need 50,000 gold and 150 hardwood.
If everything above is done a cutscene will occur when you go to bed (small chance of getting the scene) where your partner will ask if you want children. Saying yes will result in you or your partner getting pregnant or your partner filing papers for adoption (for same-sex couples). After 14 days, they will give birth and an infant will appear in the nursery. After an additional 14 days, the infant will grow into a toddler and so on. It is possible to have 2 children.
Time doesn't seem to be a factor.