Definitely! Firefox has a service, called firefox sync, that will sync all your data (passwords as well as history, settings, bookmarks, etc.) between your computers.

this support article explains how to use firefox sync: How do I set up Sync on my computer?

What Firefox version did you use on the other computer?

You can copy the logins.json and key3.db from the Firefox program on the other computer to the profile folder that you currently use.

You can use this button to go to the currently used Firefox profile folder:

• Help > Troubleshooting Information > Profile Directory: Show Folder (Linux: Open Directory; Mac: Show in Finder)
• http://kb.mozillazine.org/Profile_folder_-_Firefox

See also:

• https://support.mozilla.org/kb/Recovering+important+data+from+an+old+profile
• http://kb.mozillazine.org/Transferring_data_to_a_new_profile_-_Firefox

Thanks!

I used both solutions and they both worked.

First, I transferred the logins.json and key3.db files from my old Firefox to my new Firfox on the new MacBook Pro. Viola! It worked!

Next, I signed on to Firefox sync. That will be the end of my problems keeping current on both computers.

BTW. I discovered that both computers are running Firefox v37.0.2. The problem is apparently with Password Exporter. It can not be installed on Firefox v37.0.2, but if you have it installed before you upgrade to Firefox v37.0.2 it keeps working.

Thanks again!