Yes a vehicles alternator should always be charging the battery at idle.
When the vehicle is idling the alternator should charge at 14 volts and it should continue charging the battery at 14 volts when the engine is running faster and the vehicle is going down the road.
Alternators that are working properly should be charging at least 14 volts and if your battery light goes on or you have an alternator gauge and the gauge is showing less than 14 volts then your alternator may not be charging properly.
Of course it's normal for your alternator gauge to drop a volt or two when you turn on your blower motor or other high amp draw electronics.
But under normal conditions without anything or much of anything on at all your alternator should be outputting 14 volts.
You can use a voltmeter and touch the ground end of the voltmeter to ground and the positive end of the voltmeter to the output wire on the alternator.
There should be a bolt sticking out of the alternator that a wire is attached too and the alternator should be stamped underneath the bolt that says either battery or a + sign.
The alternator should be outputting 14 volts when you touch the voltmeter to the output of the alternator.