Do air traffic controllers notify other airports when a flight arrives?

For example, suppose that a Delta flight from Atlanta arrives without incident at Detroit. Would Detroit's ATC have to notify Atlanta's ATC of this? Or would this be a matter for the airline's operations department?

ATC is basically a relay system. When the flight leaves one sector, it is picked up by ATC in the next sector. There's no need to let the originating ATC know that the flight has landed.

It goes on the system that all relevant parties can access

They can.
There's not much need to do so.
The pilots and internal things on planes send messages to the Airline control center that the plane has landed, the plane is at the gate. And other stuff.

Far away Air traffic control is only really concerned when the plane DOES NOT arrive as expected.

For the purpose of your question The Airline operations department is the ATC of the Airline.

Delta had a big problem as well as other Airlines about twenty years ago at the Detroit Airport.

The planes landed as normal. There was a blizzard. The crews to deal with the planes were stuck in traffic. Eventually later planes were diverted to other Airports A very chaotic day for those involved.

It took a while for the control center to understand the issue that the plane had landed but was stuck getting to an empty gate.

The Air Traffic controllers care that the flight arrives safely at its destination, they don't really care how and when it gets there once it leaves their sector, as long as the flight does not end up on the evening news. The Atlanta Tower controllers hand the departing flight to the Atlanta Departure controller and the flight becomes that controller's problem, and so on and so on until the flight blocks in at its arrival gate.

Delta, on the other hand, cares very much that one of its flight arrives safely and on time. A late or diverted flight creates all sorts of problems for connecting passengers and aircraft and air crew schedulers.

Not per se. Not by phone at least. They log it into a computer which the airlines have access to. Obviously the operations center of any airline wants to know when one of their planes departs or lands or has a problem. They need this info to make sure flights are on time or to substitute a plane if needed.

Flights are passed off from one ATC to the next, and they are of little concern once they have been passed off. Arrival information is logged into a computer system, which can be accessed by the people who need that information.

If they don't they should