Hi Jackiea48,
Congratulations on your new Mac mini! We understand that Messages conversations contain a lot of valuable information, and we're disappointed to learn that you're not seeing the conversations on your new Mac. So we're on the same page, are you using the steps from Use Messages in iCloud ? If so, which other device are you trying to sync the messages from?
If the messages previously appeared on another Mac, do you have a Time Machine backup from that Mac as well? Having a Time Machine backup may allow you to restore the messages in the event that Messages in iCloud wasn't enabled before.
Let us know what you find, and we'll take it from there.
Have a great day.