The Request to Book option is a relatively new feature on Booking.com and is currently only available in certain markets. You can contact Booking.com Partner Support directly to inquire whether or not this is available to you.
At this time, Booking.com does not forward Requests to Book to Hospitable. Hospitable receives these requests only once they have become bookings.
Hospitable is, therefore, unable to show the request in your inbox or on your calendar until the booking is confirmed. If you do allow Requests to Book on Booking.com, then you'll want to keep an eye on your Extranet for those requests and reply to the guest there manually.