No, the alternator doesn't care about battery voltage. The battery can't discharge backwards through the alternator and you need a minimum engine speed for the alternator to make enough voltage, but the alternator output depends only on engine speed. If the battery is at a low level of charge, it draws a lot from the alternator and charges quickly, if it is full and has a higher voltage, close to the alternator output voltage, only a small current flows into the battery.
Just put a multimeter on the battery to see the voltage when the engine is running.
Get someone to work the throttle and see the voltage rise a little.
Be careful. 12V is not high but a car battery can provide the amps to melt your test leads if you put them in the wrong place.
Car battery charging is usually done at 13.8V (float) to 14.7V (maximum, fast charging). If you have a bench power supply these two numbers are handy to know when you want to use it to charge your car battery.