No, the Message Persistence add-on is enabled for each key so all messages are persisted for all channels published to that key by default.
NOTE: For select SDKs (Objective-C, Java, Android, JavaScript, Node.js and potentially some others), the publish method has a parameter that can be set to false to prevent the message from being persisted. For JavaScript, this parameter is called store_in_history.
Consult each SDK's docs for details on parameter name and usage.