By every eyewitness account I have ever read and by the film footage of the 1963 "Burning Monk" incident, it does appear that Thích Quảng Đức did indeed remain sitting, stoic and quiet, as he burned himself to death.

Here is a link to a video showing Thích Quảng Đức's self immolation Obviously, this video is NSFW.

And here is a good account of the incident from American journalist and photographer Malcolm Browne, who was in Vietnam at the time and witnessed the self-immolation and photographed it. This video is also NSFW, as it shows the photographs Browne took of the incident with a voiceover of him describing the rather horrific scene. In his description of the incident, Browne states that there was a clear expression of anguish on the face of Thích Quảng Đức, but that he never cried out in pain at all during the ordeal, and as the footage shows, the monk remained seated in the classic cross-legged "Lotus Position" after deliberately igniting his gasoline-soaked body with a match, until he just (to quote Browne's eyewitness account directly), "eventually he just flopped over and twitched a few times, and that was it."