No, it doesn’t work that way. If you want to update the name of the note, go to that note and rename – that way all links are updated.
I think the behavior you’re expecting would cause more trouble than benefit if you think about it, what if you want to change [[One thing]] to [[Something different]]? It will rename “One thing” to “Something different”, which is often not what you intend to. Maybe you want to change the link to something else, not updating the title of the note.
Hope that makes sense!