Hi @Parker1090. 
Hey, that’s cool! 
We use Discourse for our support inbox as well, and I’ve had various inboxes in Discourse over the years, so I’m fairly confident it is not currently possible to automate closing PMs in this way.
However, please share more details about your objective, there may be another approach.
For instance, it seems you want those discussions to definitely end at some point, but it is not apparent to me why. Let us know. 