To migrate the user alerts, Sharegate requires that the server extension is installed on the source and destination server. Here is an article how to install it.
SharePoint 2003 and Office 365
The server extension is not compatible with SharePoint 2003 or Office 365. Therefore, the migration of user alerts to or from a SharePoint 2003 server or Office 365 is not supported. Here is an article that explains what is supported and what is not for each SharePoint version.
Also, to prevent the User alerts from being triggered during the copy, the user must be farm admin at the destination.