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 events
Note is added > Private note
On tickets with these properties
--None--
Actions:
Set Status as > Open or any custom status (optional)
Send email to Agent > Assigned agent > customize your email
