
Because fullcalendar is a three-party library, I suggest you go to the library’s GitHub to ask how to add a ‘+ Add’ button to this control.
After you have added the button,you could use SP.UI.ModalDialog.showModalDialog() to create a dialog to add a new item.
SP.UI.ModalDialog.showModalDialog() document for your reference:https://learn.microsoft.com/en-us/previous-versions/office/developer/sharepoint-2010/ff410058(v=office.14)