Which country was the first to introduce this concept?
Well the concept of government providing healthcare dates back to ancient times, even though the healthcare itself would hardly have been something that we'd recognize as valid treatment today. The earliest known record that I can think of that's actually documented is from the Rock Edicts of Ashoka Maurya(r.269-232BCE, India, Modern day Patna).
"Everywhere within Beloved-Servant-of-the-Gods, King Piyadasi’s domain, and among the people beyond the borders, the Cholas, the Pandyas, the Satiyaputras, the Keralaputras, as far as Tamraparni and where the Greek king Antiochos rules, and among the kings who are neighbors of Antiochos, everywhere has Beloved-Servant-of-the-Gods, King Piyadasi, made provision for two types of medical treatment: medical treatment for humans and medical treatment for animals. Wherever medical herbs suitable for humans or animals are not available, I have had them imported and grown. Wherever medical roots or fruits are not available I have had them imported and grown. Along roads I have had wells dug and trees planted for the benefit of humans and animals. Rock Edict Nb2 (S. Dhammika)"
Ashoka is famous for being a ruler who was concerned above all for the health and well being of his people, though how much of that is true and how much is propaganda from himself and the Buddhist writers who revered him for being the first patron of Buddhism is unclear.
The notion in the modern world, in a form that we'd recognize today as a government healthcare program, first came about in Germany, right around the first time that Germany was first a thing, Health Insurance Bill of 1883, Accident Insurance Bill of 1884, and Old Age and Disability Insurance Bill of 1889 being the primary examples. Bismarck was himself a tremendous proponent of the idea of healthcare provided by the state for, in particular, lower income laborers, and the system was pretty quickly expanded to become the precursor of many modern universal healthcare systems.