You have three ways to do this:

On ES5:

1. Add .bind(this) at the end of the function, to be able to access this inside the function

   channel.bind('App\\Events\\UserHasNewMessage', function(data) {
        this.messages.push(data);
   }.bind(this));
   

2. Assign this to a variable, and use that variable inside the function

   var that = this;
   channel.bind('App\\Events\\UserHasNewMessage', function(data) {
        that.messages.push(data);
   }.bind(this));
   

On ES6:

3. Use the fat arrow, so you can use this inside the function

   channel.bind('App\\Events\\UserHasNewMessage', (data) => {
        this.messages.push(data);
   });