My mother was a Towey from Ballaghaderreen so your question caught my eye. It seems the answer is no. I believe Toohig/Twohig (and similar) are derivations of the old Cork name O'Tuathaigh. ie descendant of Tuathaigh (or Tuathach), which meant chief, lord, over tuath ('tribe', 'territory'). Towey, Toohey etc derive from the old Mayo name of O'Toghdha, ie descendant of Toghdha, (toghdha - 'chosen'). I will gladly stand corrected.