I believe, and others may correct me if I’m wrong, but the only time you “know” that someone has purchased something from a quickshop is by the message in chat when the transaction happens, or alternatively, if you’re offline, you’ll see it when you next log in.
So you could make your own excel spreadsheet and write these down as you see them happen, but as far as I know, there is no /command that will show you a history of purchases.
I hope that answers your question.