Freshdesk does not have a default agent email notification for notes added by a third party (reply to a ticket forward). Instead, you can configure an automation rule under Admin > Workflows > Automations > Ticket updates > New Rule:

When an action performed by...

Agent or requester

Involved any of these eventsĀ 

Note is added > Private noteĀ 

On tickets with these properties



Set Status as > Open or any custom status (optional)

Send email to Agent > Assigned agent > customize your email