Whenever a NPC is talking, the game fires a EVENT_SHOW_SUBTITLE. The subtitle manager listens to this event and displays the subtitle whenever the event is fired (see line 87 of your linked source).
The subtitle manager displays the subtitle by calling OnShowSubtitle, so I'd call that function:
Lua Code:
ZO_SUBTITLE_MANAGER:OnShowSubtitle(0,"Name","message")
The first argument is the message type. I don't know what types there are, but 0 worked fine when i tested it,