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.
The Server extension is only required for On-Premises environments, not Office 365.
The server extension is not compatible with SharePoint 2003. Therefore, the migration of user alerts to or from a SharePoint 2003 server is not supported. Here is an article that explains what is supported and what is not for each SharePoint version.
To prevent the User alerts from being triggered during a copy to an on-premises environment, the user must be farm admin at the destination.
When copying to an Office 365 environment, there is no way to prevent user alerts from being triggered. Since it is a migration to a cloud environment, Sharegate does not have the information on when certain tasks are completed, which may results in alerts being triggered during the copy.