That blue-background element is not a Contact Form 7, that is a Classic Callout element. The purpose of the button in there is to link to your actual Booking page.
Now on your actual Booking page, you can use Contact Form 7 plugin for creating a form, but that plugin is not enough for booking because that is only a contact form, you will need some add-ons for it to work as booking plugin.
I would advise that to use a plugin that the purpose is for booking, instead of Contact Form 7 unless you just want the Booking to be sent to your email.
You can try the Modern Event Calendar extension, this has a booking feature.