Bosola has the Duchess eat the apricots to confirm his suspicions that she is pregnant. Her pregnancy is confirmed when she throws up and goes into labor. It's suggested that they should say the apricots are poisoned so no one will suspect that the Duchess is giving birth.
I think this scene reflects the time period in which it was written. Knowledge of pregnancy and how pregnancy affected women was not great, so it's assumed that the Duchess will throw up (morning sickness?) even though most women suffer from nausea during the first few months of pregnancy, not at the end. Also, Bosola sees the sickness of the Duchess as proof positive of pregnancy, when in fact, throwing up doesn't always mean you're pregnant. Some women don't even crave certain things when they're pregnant.