Laravel 5 controller to list appointments by date range and by label
The structure looks ok to me (I'm not that familiar with laravel, but this is basically how MVC in a web context generally looks). A couple of minor points:


you don't really need one-time variables such as $appointments, just return directly.
findAppointments* could be renamed to find*. It's in the Appointment class, so it's expected that it will find an appointment, unless noted otherwise. This also avoids the odd phrasing of Appointment::findAppointments and makes it Appointment::find.
be as explicit with your variable names as possible (here, it's not so important, but it's good practice, and for larger methods it does get important). eg value could be label, or even labelId, and start/end could be startDate/endDate (it may seem that this is a contradiction with the point above, but it's really not; the variables will be used inside the method, missing the direct context of the method name, while the methods will be called either directly on the class as you are doing, or on a variable, which will be named after the class, giving them direct context). 
You can use self:: instead of ClassName::, even in static methods.