Message's transfer to new iMac

You can sync your iMessages in iCloud. On both of your iMacs, open Messages. Click Messages in the menu bar, then Preferences. Click iCloud, then make sure that Enable Messages in iCloud is checked.

You can get step-by-step instructions here: Keep all your messages in iCloud - Apple Support

If you don't see the option to enable Messages in iCloud, make sure that your iMacs are up to date. Messages in iCloud requires macOS 10.13.5 or newer.

As mentioned the only method I know works for sure is to use Migration Assistant. Did you launch the Migration Assistant app when you migrated from your old Mac to the new one? If not you still can but it will require that you start over and erase your new iMac and just install Mac OS.

If all you care about is your Messages content then I'd recommend that you call AppleCare and see if they can help you out.

AppleCare contact Information

If you use Migration Assistant to migrate your data, your messages will come over.