Q: My maternal grandfather married my paternal cousin. Thus, my paternal cousin, the wife of my grandfather, became my mother's aunt (stepmother).
Is it permissible for me to shake hands with my paternal cousin who married my grandfather and became my mother's aunt?
May Allah reward you with the best for serving Islam and Muslims.
If the reality is as you have mentioned, being the wife of your maternal grandfather renders your paternal cousin a Mahram (unmarriageable relative) to you. Thus, you may shake hands with her. May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.