They met only once, briefly and by accident. They both happened to be in the waiting room of the Secretary for War, Lord Castlereagh, on 12th September, 1805.
Wellington described the meeting 30 years later. He said that at first they hadn't recognised each other. "He [Nelson] entered at once into conversation with me, if I can call it conversation, for it was almost all on his side and all about himself and, in reality, a style so vain and so silly as to surprise and almost disgust me."
After a minute or two, Wellington noticed that he had only one arm, and wondered if this could be Nelson. Then Nelson excused himself and left the room for a minute. He must have asked someone who that chap in the waiting room was, and been told it was General Arthur Wellesley.
When Nelson came back he spoke in a completely different way, and they had an intelligent and interesting conversation about politics and the war, lasting about 15 minutes, until one of them was called in to see the Secretary for War.
Nelson was killed at the battle of Trafalgar just over six weeks later.