You can use the following information when deploying a Node.JS application via cPanel's Application Manager to view the output of console.log that you've added to your application code.
The mod_passenger module that cPanel makes use of will publish the output from console.log to the Apache error log. In order to view your console.log output, you'll need to make sure that your application is configured properly so that mod_passenger is actually executing your application code. We have a blog post here about setting up a test Node.JS application:
Once you are confident that your application is setup properly, deployed, and your code has a console.log call, all you would have to do is tail the Apache log with the following command via SSH as the root user:
tail -fn0 /etc/apache2/logs/error_log
Then, visit the domain that your Node.JS application is setup on to make mod_passenger execute your code and print to the Apache log.