is the charge light lit on the dash with the engine running?
check the main output connection on the alternator to the body of the alternator with a volt meter, if it reads <13v then yes, the alternator is probably dead
ah, sorry, i thought you meant the battery was at 12.3v while running.
take the battery off and put in on charge for 24h, then leave it for an hour and check again. if its still saying 12.3v then take it back to the shop
are you getting 14.7v at the battery terminals when the engine is running? If not then you need to check the lead from the alternator to the battery +ve and engine earth.
the alternator isn't designed to fully charge a flat battery, it needs to be fully charged to start with.
if you take the battery off (disconnect) and make sure its fully charged, you have a base to start from.
how old was the previous battery?
did you check the batteries when they were new, before you put them on the car? dare i ask the retailer? and did they check the old battery before replacing it?
Battery test should include CCA / EN info to properly indicate condition. Static voltage simply indicates state of charge at the time.
As Kermit says you need to check voltage at alternator and battery to get an indication of charging. Using an ammeter also helps, but with the engine running measuring at the battery cables gives the combination of charge current minus the load from engine & other ECU's. Alternatively current in the alternator cable give charge current plus all the ECU's, etc. Measuring both ends can give a good indication of the charging.