How can I monitor if the Laravel queue is running?

Actually, when a queue fails a failing event fires, so for example, you may register the failing event in your AppServiceProvider class in the boot method using something like this:

public function boot()
{
    Queue::failing(function (JobFailed $event) {
        // $event->connectionName
        // $event->job
        // $event->data
    });
}

Alternatively, you can declare a failed method in the handler class for example:

class SendEmail extends Job implements ShouldQueue
{
    use InteractsWithQueue, SerializesModels;

    public function handle(Mailer $mailer)
    {
        //...
    }


    public function failed()
    {
        //...
    }
}

The Updated link.

Regarding background monitoring, you may use Supervisord:

Supervisor is a client/server system that allows its users to monitor and control a number of processes on UNIX-like operating systems.

In this case, you have to install it on your machine and configure it using at least one program section, for example:

[program:queue]
command=/usr/local/bin/php artisan queue:listen --tries=3

This is an example of program section that I've used to monitor my queue using Supervisord. In this case, you need to read the documentation for the supervisord to understand how to use it, I've just gave you an idea. The supervisord will run in the background once you start it and it'll also restart the observation even after the server is restarted (if it goes down for some reason) so you don't need to worry about that.

A simple (minimal) config file may look something like this:

[unix_http_server]
file=/tmp/supervisor.sock   ; (the path to the socket file)

[supervisord]
logfile=/home/someDirName/www/example.com/supervisord.log
logfile_maxbytes=50MB
logfile_backups=10
loglevel=info
pidfile=/tmp/supervisord.pid
nodaemon=false
loglevel=warn

[rpcinterface:supervisor]
supervisor.rpcinterface_factory = supervisor.rpcinterface:make_main_rpcinterface

[supervisorctl]
serverurl=unix:///tmp/supervisor.sock

[program:queue]
command=/usr/local/bin/php artisan queue:listen --tries=3
directory=/home/someDirName/www/example.com
autostart=true
autorestart=true
redirect_stderr=true

Well, you may read the documentation to really get the clear idea about it. This may help you to start.

This is lower level but in the same vein you could run a command such as ps -aux | grep queue to literally view the running queue processes on whatever server your application/queue workers are running on.