I find this really annoying and would like to be able to disable it.

It has nothing to do with them, yet pops open, when the other  new quests complete, and if its closed it, pops open again on claiming these, and remains there till its closed.

I do not need to be constantly reminded that I have unclaimed items.

Nor accidentally claim them when attempting to shut the banner instead of claiming the new quest, which has its own alert anyway.