Hi! To disable outgoing HivePress emails, go to WordPress Dashboard->HivePress->Emails, and add a new email. scroll down, under the editor window, select an event for that email. Leave the email content empty, and save. Do it for every events. Because the system will not send an empty email, your HivePress will not send more emails. If so want to edit the HivePress emails, with this method, you can customize every emails.
Take a look here: How to set up emails
And here: Email template
Ps.: You need an SMTP plugin to be installed in your wordpress (WP Mail SMTP or similar)