On trying to move a department from within the Departments page via drag and drop, the error message ''Cannot insert duplicate key row in object 'dbo.department' with unique index 'uix2dep_parentname'. The statement has been terminated." error is thrown as a pop up. (Refer to the Screenshot attached.)

There is a department with the same name as the department that you are trying to move in the same hierarchy. Web TimeSheet does not support having 2 departments at the same hierarchy level with the same name. The department names in the same hierarchy level has to be unique.
Either you can rename the original department which exists in the same hierarchy to differentiate it or rename the department that you are moving so that you can have both the departments in the same hierarchy.