How harmonics happens when you mix two signals?
Amplitude modulation is a linear operation (excluding the carrier in the classical AM) and does not introduce harmonics due to the modulated signal. Sidebands are not harmonics. They are created as the message signal spectrum is shifted up to carrier frequency.
Frequency mixing of a message signal with a sinusoidal carrier, another definition of AM, does not introduce harmonics.
In a practical setting if there are some nonlinearities in the implementation (a deviation from the ideal case) you may have harmonics. Otherwise No.