Facebook has a feature that allows business page administrators to add a configurable button to their Facebook page. This can be used to allow clients to make bookings directly from Facebook. Here's how:
Click on the 'Create Call-to-action' button. Note, you'll need to have a Facebook business page and logged in as the administrator.
Enter your business' public page booking URL and click Next
Voila! That's it. Make sure to check that your button works as expected by clicking on it and choosing "Go to link". Of course, visitors will not see multiple options like you do when clicking this button - they'll go straight to your booking page.