I don't think there is.
You could set it without sending a message: start to compose a new message to the person, then return to the conversation list without sending. You now have a new conversation with a 'draft' message. If you return to the conversation you can now customise the notification for it the same as any other. However, what I don't know, because I've not tested, is whether it will remember the custom notification if you delete the draft message. I suspect that when the final (only) message is cleared it will forget everything related to that conversation, because there will no longer be a conversation. If so you could do it, but only by cluttering your conversations list with junk texts until you have a real exchange with that person (at which point you can lose the drafts). For me that wouldn't be worth it, but that's just me.