For anyone who would like to use Google API with a service account connection, here is what I have to done in order to make it work :

1- Call the following in command line : composer require google/apiclient

2- In your composer.json file add the following :

{
    "autoload": {
        "classmap": [
            "vendor/google/apiclient/src/Google"
        ],
    }
}

3- In your http://console.developers.google.com Create a new Client ID in Credentials, Select Service account and click on Create Client ID

4- Next, take the *.p12 file and place it somewhere in your laravel directory (in my case in assets).

5- Next, create a new Gooogle calendar (http://calendar.google.com) with any Google account and share it with your Service Account's Email Address created in step 3 and save it.

6- In the config folder, add a file name google.php, place and modify the following content :

<?php

return [

    /*
    |--------------------------------------------------------------------------
    | Client ID
    |--------------------------------------------------------------------------
    |
    | The Client ID can be found in the OAuth Credentials under Service Account
    |
    */
    'client_id' => 'something.apps.googleusercontent.com',

    /*
    |--------------------------------------------------------------------------
    | Service account name
    |--------------------------------------------------------------------------
    |
    | The Service account name is the Email Address that can be found in the
    | OAuth Credentials under Service Account
    |
    */
    'service_account_name' => 'something@developer.gserviceaccount.com',

    /*
    |--------------------------------------------------------------------------
    | Key file location
    |--------------------------------------------------------------------------
    |
    | This is the location of the .p12 file from the Laravel root directory
    |
    */
    'key_file_location' => '/resources/assets/filename.p12',
];

7- In your application folder, I have added a file named GoogleCalendar.php in App\Services with the following content :

<?php namespace App\Services;

use Illuminate\Support\Facades\Cache;
use Illuminate\Support\Facades\Config;

class GoogleCalendar {

    protected $client;

    protected $service;

    function __construct() {
        /* Get config variables */
        $client_id = Config::get('google.client_id');
        $service_account_name = Config::get('google.service_account_name');
        $key_file_location = base_path() . Config::get('google.key_file_location');

        $this->client = new \Google_Client();
        $this->client->setApplicationName("Your Application Name");
        $this->service = new \Google_Service_Calendar($this->client);

        /* If we have an access token */
        if (Cache::has('service_token')) {
          $this->client->setAccessToken(Cache::get('service_token'));
        }

        $key = file_get_contents($key_file_location);
        /* Add the scopes you need */
        $scopes = array('https://www.googleapis.com/auth/calendar');
        $cred = new \Google_Auth_AssertionCredentials(
            $service_account_name,
            $scopes,
            $key
        );

        $this->client->setAssertionCredentials($cred);
        if ($this->client->getAuth()->isAccessTokenExpired()) {
          $this->client->getAuth()->refreshTokenWithAssertion($cred);
        }
        Cache::forever('service_token', $this->client->getAccessToken());
    }

    public function get($calendarId)
    {
        $results = $this->service->calendars->get($calendarId);
        dd($results);
    }
}

8- Next add the following where you need to use Google Calendar

use App\Services\GoogleCalendar;
[...]
    public function functionName(GoogleCalendar $calendar)
    {
        $calendarId = "YourCalendarID";
        $result = $calendar->get($calendarId);
    }