I guess that you don't want the two components to know about each other.
I suggest you to have an other class (let's call it glue) with references to both the quest and the item managers. This class will be observer of the quest manager (google for the observer pattern *), and will be responsible to call the required method of the item manager.
* If you look here you see a C# example using the events.
In your case you first define a QuestCompletedEvent and add a field of that type in the Quest class. Then you subscribe the Glue class to it, by creating a method OnQuestCompleted which takes the senter and the arguments (as shown in the example) and subscribing that method with the += operator (see example). Secondly, inside the logic of the Quest class you add
a .Notify() when the quest is completed, this way the glue Class gets
notified, i.e. the OnQuestCompleted method is called. Inside that method you will run the code necessary to update the item manager.
I hope this clarifies the thing. The only thing you need then is to find a proper name for the Glue class ;)