If I delete a queue, will this also delete all the chats associated with that queue?

No.  Deleting a queue will not delete the chats associated with that queue.  Chats on deleted queues will continue to be visible and searchable in the Chat History area.  

The caveat is Reports.  A deleted queue will no longer appear in the list of queues for selection, and so chats on deleted queues won't be included since you'll usually select the desired queues from that provided list.  If you need to run a report that will include chats on deleted queues, the trick is to unselect all of the queues from the list of queues in the report module.  Then, the report will run across ALL queues, even the ones that have been deleted.

FAQ URL: