Yes, he did: Churchill gave the speech in Westminster Hall to a joint meeting of the Lords and Commons at the unveiling of a portrait of him painted by Graham Sutherland for his eightieth birthday. With tremendous ambiguity, he described it as "a remarkable example of modern art. It certainly combines force and candour."

As shown in the episode, Graham Sutherland had a reputation for refusing to flatter his subjects, which some regarded as cruel or disparaging. Churchill was not taken by surprise by the portrait, having been shown a photograph of it by his wife Clementine beforehand. He was, however, enraged by its depiction of him. Clementine recalled in 1957, ‘It wounded him deeply that this brilliant... painter with whom he had made friends while sitting for him should see him as a gross & cruel monster.’ He reportedly almost shouted at his solicitor, Anthony Moir, questioning whether or not it was a libel. However, Members of Parliament had paid for Sutherland's 1,000-guinea fee out of their own voluntary donations, so there was no question that the portrait would be accepted.

Despite what the episode implies, Churchill did not resign immediately after the portrait incident in November 1954, holding on until the next April. The portrait's fate is portrayed relatively accurately: it was consigned to the cellar at Chartwell, all requests by galleries to borrow the painting for exhibitions were refused, and after Clementine Churchill's death in 1977, it became clear that she had had it burned.

Sources:

Andrew Roberts, Churchill: Walking With Destiny

Hugo Vickers, The Crown: Truth and Fiction