Does the Samsung Galaxy S4 support HDVoice ?

For information on HD Voice, including capable devices and current HD Voice coverage areas, please visit att.com/hdvoice  And don't forget to click on the FAQs!

I know the S4 Mini does. So, that might imply the S4 does too.

It does, but enabling it is non-trivial. If you google for Galaxy S4 HDVoice you will find several sites walking you through the specific sequence in the dialer to get to the setting to enable it.