Yes, you can check the voltage from the positive battery terminal to ground. Probably around 12.5 to 13 volts with the engine off, then 14 to 15 volts when it's running. If there is no change in the battery voltage when you start the engine, then the alternator is not charging.
You say there are "no wires on the alternator". If, indeed, it has never been wired up then it's not going to be charging the battery.