Give a service a minimum notice, e.g. three hours, and clients can no longer request a time that starts sooner than that.
Prerequisites
Go to Appointments → Appointment Types, find the service, and click Edit.
Pick how much warning a client must give — from 1 hour up to 2 days. "No minimum" is the default and keeps today's behaviour.
Click Save. Times inside the window stop appearing for clients straight away, and the booking page explains why: "Requests need at least 3 hours notice."