how to transfer just the mailbox from time machine to a new Mac

What exactly do you mean by "just the mailbox"? The Inbox? If so, I don't think that can be done. i.e. a single mail folder such as Inbox, Drafts and etc.

If you run Migration Assistant, your mail boxes will be migrated. After that you can remove any account or folder you wish.

Or you can create a mail account from scratch and remove what you don't want. Mail is little more than a local app that allows local control of a web based mail account. As long as you didn't touch the web account, all your emails and folders are still there.