Note that the frequency response
is real-valued and even, and, consequently, the corresponding impulse response is also real-valued and even, i.e., it's a zero-phase type I FIR filter.
The frequency response $(1)$ can also be written in the form
So who is right?
Both, I think. On first looks version  and version  use different definitions of $A(z)$.  conjugates the zeros and  conjugates the poles. Either one will probably work but the definition of the coefficients is different. Specifically the $a$ coefficients of version  or conjugates of those of version .
So you have
Typically a transfer function is described as a function of the complex variable $s$ as given by the Laplace Transform of the systems impulse response:
By expressing the transfer function of a linear system as a ratio of two such polynomials, we are able to describe the linear system uniquely in terms of the roots of those ...
I don't think they get a special name -just the numerator and denominator function.
Notice that the frequency response is defined as the ratio of output spectrum to input spectrum, but since the input signal is called X and the output Y, I don't think N and Q have a direct relationship with them (lest you're omitting some context from the material where you ...