Hey man, welcome to A2K.
No rule like you talkin' about exists.
At this linkare the NFL rules for substitutions, and they don't mention anything about specific QB substitutions.
Let's address your scenario, however:
If a QB (or any other position player, for that matter, offense or defense) is injured seriously enough for his team to be forced to call a timeout in order for him to receive treatment or be assisted from the field
in the last two minutes of each half, then that player cannot remain on the fieldfor the next play from scrimmage; he
must leave (for at least one play).
So if a team's starter (at the end of the second or fourth quarter) got sacked, lay on the field writhing, and a timeout was called, he has to go out. And if the second-string QB, while running to the huddle, sprains his ankle, and can't get off the field, forcing his team to call
another timeout, then the third-stringer
has to come in.
But only for one snap.
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