Hi everyone and thanks for taking the time to share your feedback with us!
I investigated this further and I confirm “Messages I’ve received” is currently not available via quick search. However, if you want to have this list available in your sidebar, you can create a report using the advanced search. You just need to add yourself as a collaborator in the search parameters, click search and switch to the messages tab.
Then you can favorite this search, so it appears in your sidebar.
Alternatively, you can also quickly access this search from this link.
I hope this helps! I will keep this thread opened and will let you know if we add this as a quick search in the future. Thanks again!