That is correct I think. It was done out of the public eye.
Vercingetorix was strangled in the Tullianum, a subterranean chamber in the Carcer, accessible only by a hole through which he was lowered by the executioner who would have gone down and then strangled him. He was tired of living and was probably looking forward to it. Disgraced, defeated and paraded like an animal, a proud man like him must have had a hard time in captivity after being king to hundreds of thousands of warriors.
This doesn't make a whole lot of sense: HBO has done a pretty good job of keeping in line with historically accuracy, so why break the trend with strangling Vercingetorix out in public, when, as is apparently the case, he was killed in secret? For all intents and purposes, he was essentially written off in the first episode after he surrendered to Caesar.
Wat up wit dis????