If your Mac Address Book is a vCard (.vcf) file, it can be imported to TB with the help of this add-on:

https://freeshell.de//~kaosmos/morecols-en.html

http://chrisramsden.vfast.co.uk/3_How_to_install_Add-ons_in_Thunderbird.html

http://kb.mozillazine.org/Import_from_Mail.app#Import_the_address_books

In the TB Address Book, Tools/MoreFunctionsForAddressBook/Actions for contacts/Import vCard/vcf.

Thank you for responding! However, that add on is not compatible with TB 31.1 So it didn't work.

I meant Firefox 31.

The add-on is compatible with TB 31+, not Firefox. See the 2nd link in my reply for instructions on how to install add-ons.

I followed the instructions in the 2nd link. In TB, I went to Tool>add ons and found the download in my download folder. In TB, I clicked Install Add-on from File. Here is the message that I get: MoreFunctionsForAddressBook could not be installed because it is not compatible with Thunderbire 31.1.0

Did you try to install version 0.7?

https://freeshell.de//~kaosmos/morefunctionsforAB-TB3-0.7.xpi

yes I tried to install v 7 and it didn't work. I also tried a Safe Start up, but that made no difference. Is there a work around for this?

I just installed MFFAB 0.7 in a test profile of TB 31.1.1 with no problems, but on Win 7. Perhaps there is an issue installing it on the Mac OS, in which case I would contact the add-on author. Almost all add-ons can be installed on Win, OSX or Linux.

Otherwise, what happens if you open TB Address Book, Tools/Import, select Address Books, then vCard file, then select the .vcf file created from Mail app?