Nope, as an Irish person 'of the South' neither I, nor anyone I've ever met, including my rather large extended family, has ever referred to Ireland as 'Southern Ireland'. If that's what you call our country, go for it. But don't tell me what 'people of the South' call it, as you clearly haven't a clue.
And yes, I'm well aware that Northern Ireland (an actual country) is part of the UK. But go raibh maith agat for the education.