stdout, when going to a tty (terminal) is an instance of a writable stream. The same is true of stderr, to which node writes error messages. Those streams don't have message events. The on() method allows solicitation of any named event, even those that will never fire.
Your requirement is not clear from your question. If you want to intercept and inspect console.log operations, you can pipe stdout to some other stream. Similarly, you can pipe stderr to some other stream to intercept and inspect errors.
Or, in a burst of ugliness and poor maintainability, you can redefine the console.log and console.error functions to something that does what you need.
It sounds like you want to buffer up the material written to the console, and then return it to an http client in response to a GET operation. To do that you would either
stop using console.log for that output, and switch over to a high-quality logging npm package like winston.
redefine console.log (and possibly console.error) to save its output in some kind of simple express-app-scope data structure, perhaps an array of strings. Then implement your GET to read that array of strings, format it, and return it.
My first suggestion is more scalable.
By the way, please consider the security implications of making your console log available to malicious strangers.
console.logbut this event still doesn't do anything.