In the book Quantum Mechanics, Volume 1, written by Albert Messiah, page no. 142-143, the author says: 


  but the diaphragm is a quantum object, just like the electron. Its momentum is not defined to better than $dp$


I do not understand why diaphragm is a quantum object.

Also, it is not clear to me what author says below:


  One must postulate that the measuring apparatus is a quantized object which also obeys uncertainty relations


On the other hand, N. Bohr says that the measuring apparatus is always a classical object (i.e. an instrument which follows the rules of classical mechanics).

Any clear explanation?

It's obviously not a sharp cut-off, but as a general guide sound waves cannot propagate if their wavelength is equal to or less than the mean free path of the gas molecules. This means that even for arbitrarily low pressures sound will still propagate provided the wavelength is long enough. Possibly this is stretching a point, but even in interstellar gas clouds sound waves (more precisely shock waves) will propagate, but their length scale is on the order of light years.