Depending on the configuration, you will likely need to reissue your Blesta license to the new domain/location.
If the credentials are all the same.
There is a README file included with Blesta that describes how to upgrade. The upgrade documentation is also useful.
Here's what I would recommend in your case:
BACKUP YOUR BLESTA INSTALLATION AND DATABASE (especially /config/blesta.php)
Put Blesta in maintenance mode
Disable your Blesta cron job
Download and upgrade to v4.0.0 as an intermediate step. This is mainly due to there being a lot of changes between v3.6.2 and v4.4.2 to make in one big leap. I would recommend upgrading via CLI mode (instead of /admin/upgrade in your browser) so that the upgrade process cannot time out in the web browser.
Upload the contents of the "blesta" directory
The upgrade documentation will be helpful to review. There are files you should delete as well, mentioned in the note at the top of that page. Some people have problems if those files remain after upgrade while most others do not.
Run the upgrade script (via CLI is preferred)
Check your Blesta installation to make sure it is working normally on v4.0.0
Backup your Blesta installation and database again, now for v4.0.0
(You can perform the server/cPanel migration here, or later at step #12-13)
Upgrade your server to use php 7.2 (php 7.2+ is only supported on v4.4.0+ of Blesta)
Ensure php 7.2 is installed for both CLI and web
Download and upgrade to the latest version (currently v4.4.2) - via CLI mode is recommended over /admin/upgrade in your browser
Since you are on php 7.2, you will need to overwrite a few files from the "blesta" directory with all the files in the "hotfix-php71" directory. You will receive ionCube errors if you do not do this
Upload the contents of the "blesta" directory, which now contains the overwritten files from the "hotfix-php71" directory
Run the upgrade script (via CLI is preferred)
Check your Blesta installation to make sure it is working normally on v4.4.2+
Update configuration settings, like cron jobs and email templates, if necessary, since many changes were made since v3.6.2
Backup your Blesta installation and database again, now, v4.4.2+
Perform your server/cPanel migration as necessary
Be aware that changes to your Blesta location (e.g. by domain or subdomain) may necessitate that you update any third-party configuration to point to the new location. e.g. PayPal IPN URL
Log in to your account with us, or your reseller, and reissue the license
Click to "Manage" the service representing your Blesta license, check the "Reissue License" checkbox, and submit
Visit your Blesta installation at your domain and log in. This will complete the license reissue at your new domain
Re-enable the Blesta cron job and observe that it runs tasks as shown on the Settings > Automation page
You may need to reconfigure the cron job to run at the new location if it has changed
Disable maintenance mode