Is Senator Alan Simpson’s quote – "a sparrow's belch in the midst of a typhoon" in comparing budget cuts frequently-used quote by Americans?

Solution 1:

A sparrow's belch in the midst of a typhoon

is not a commonly-used idiom; I do not think it is standard, either, but its meaning can be easily deduced. English speakers routinely create their own metaphors in place of well-worn idioms. This phrase may well be Simpson's own coinage, as he used it almost exactly five years ago in an interview with Fox News Sunday:

Let me tell you, those who don’t like him have put a big red tail on his bum, and cloven hooves, and horns on his head. And let me tell you, if anybody thinks — if this had happened to anybody else in America, it would have been like a sparrow belch in a typhoon.

A more common standard equivalent is:

A drop in the bucket/ocean

Solution 2:

Alan Simpson was probably giving a euphemism for the much more common

... like a squirrel fart in a thunderstorm ...

Google squirrel fart if you don't believe me. There's even video. :)

Simpson is a politician, and a Republican to boot, so he doesn't want to offend the electorate with "crude" language, especially anything that might be construed by some old lady in Dubuque to be scatological or otherwise offensive.