Hello Richard,
Thanks for reaching out!
I believe that would be absolutely possible with Events Calendar PRO using categories for the different departments.
The main calendar page (usually accessible at domain.com/events, but that can be customized) would show the events of all the departments.
Separate pages could be set up for each department where the calendar could be included with a shortcode filtering to a specific category (department). Or you can also have them without separate pages simply using a url like this: domain.com/events/category/{department-name}
I believe with a bit of custom code you could even change the “category” slug to “department” if needed.
And, though not developed, maintained or supported by us, The Events Calendar Category Colors plugin would be a nice addition, so you can differentiate the different department events on the main calendar.
Does this answer your questions? Please let me know.
Cheers,
Andras