The #DEAD_MSG_QUEUE will show up in the web console just like any other queue. You’ll see its message count.
In order for a message to be moved to the DMQ, the message needs to be DMQ eligible and the Maximum Redelivery Count configured on the queue needs to be exceeded. See DMQ docs
If your operation is a LISTEN, it needs to be configured as either Persistent Non-Transacted, or Persistent Transacted.