It would seem that a faulty alternator couldn't cause transmission problems.
But indeed a faulty alternator could cause problems with your automatic transmission since automatic transmissions have electrical sensors and when the alternator doesn't send the correct current to the transmission sensors such as shift sensors or other solenoids then yes the transmission could have shifting problems due to a faulty alternator.
Excessive AC interference from a faulty alternator causes the computer to also misread your vehicles speed and RPM which messes with the transmission shifting.
A bad battery can also cause shifting problems with your transmission.
If your transmission is full of fluid and the fluid is clean and your transmission filter is not dirty then the alternator or battery could be at fault.
Also the sensors could be bad or faulty as well.