When did the public opinion started to question the propaganda lies?

You have to remember that to do so was a crime that carried a potential death sentence. Assessing something as broad as 'public opinion' under conditions of such draconian suppression is a bit ropy. One must speculate to some degree, or narrow the inquiry.

That said, he regime was highly sensitive to the public mood; probably the best source for this answer would be Goebbels himself (via his reasonably frank diary), and the changing tone of propaganda over time. As late as September 27 1941 Goebbels complained of "excessive optimism", a mood summed-up by a Hitler speech on October 3 in which declared victory as good as won, and this set the tone for propaganda. Newspapers declared "Campaign in the East Decided!", and cinemas advertised a forthcoming documentary titled The Germans Enter Moscow. Of course, as we know, this was all slightly premature, and Goebbels worried about a "psychological defeat" if further military progress failed to eventuate (he knew the army was not in great shape, but was reassured by Hitler's fanatical optimism). A month later, winter was falling, with neither Moscow captured, or the Red Army defeated, and the army was in dreadful shape - Goebbels was aware that, despite their best efforts (increased censorship of field post, illicit foreign broadcasts, and suppression of things like death notices), the falsity of their previous claims would become known one way or another, and to try to say otherwise would damage the regime's credibility. Propaganda hereby shifted from triumphalist proclamations of imminent victory, to guarded optimism coupled with exhortations to greater effort and sacrifice, and renewed fearmongering about the fate of Germany if they lost in the east. It seems likely that even the most ill-informed civilian must have sensed something was going wrong simply by listening to official propaganda contradict itself, which is borne-out by significant anecdotal evidence from letters and diaries (particularly from well-connected elites and the military).

So, the answer: between October and November 1941.