have him create a user account for you on his Mac.
Log into your account on the Mac and sign into your iCloud account.
Turn on Messages in iCloud on your phone if it's not already on.
Turn on Messages in iCloud on the Mac.
All of the message conversations on the phone will sync to the cloud, and down to the Mac. (this might take some time).
Once they do, open messages on the Mac, click on the conversation in the left hand panel of the messages window to hi light it, then click on the file menu and click print. That will print that conversation thread to paper or to a PDF file.