not having a battery light dosent mean anything. its really more of a dummy light. if you have no volts at the alt then it is time for a new one. better get it fixed fast low voltage will cause FICM failure and that is not a cheap part to buy.
Just as an FYI, low battery voltage can damage the FICM which can turn an alternator or battery issue into a much more costly repair. I would recommend that you get the problem diagnosed and fixed quickly before it becomes a major issue.
While the alternator is out I would have both batteries individually load tested to make sure they are also up to snuff. If one is bad replace both. The other won't be far behind and can drag a new battery down.