Acuity doesn't have a built in automation feature, but if you're interested in sending clients a birthday email or coupon for their birthday, we do have a workaround.
You can include a birth date field in an Intake Form and additionally make that field required for clients when booking an appointment. You can use the Textbox field to gather this info:
Next, head to Appointments > Reports > Export. Tick the box next to the intake form with your birthday question, then export. You can then import that list into an email marketing program that has automation.