Is it possible that my band size will increase post Breast Augmentation?

This will depend on how you want your bras to fit.  Theoretically the band size will not increase because it is a measure of your chest circumference.  However, with implants you might find the 34 is mor comfortable.  It will require trying the bras on.

This is a function of your rib cage or thorax below the breasts and therefore should not be expected to change.