Let $R$ be an $\mathbb{N}$-graded ring with $R_0$ Artinian and $R = R_0[x_1,\dots,x_r]$, where the degree of $x_i$ is $d_i > 0$. Let $M$ a finitely generated $\mathbb{N}$-graded $R$-module with Hilbert function $H(M,n)$. Then it is known that there exists a unique polynomial $P_M(t)$ such that $H(M,n) = P_M(n)$ for large enough $n$. This result can be found e.g. in Matsumura's Commutative Ring Theory at pages 94-95.
In Bruns&Herzog Cohen-Macaulay Rings, a quasi-polynomial is defined
to be a function $f:\mathbb{Z} \rightarrow \mathbb{C}$, such that $f$
is a periodic piecewise polynomial. Then Theorem 4.4.3 reads as follows:
Question: I am failing to see in what way the setting of this theorem is a generalization of the setting described in the first paragraph of this question above. This has to be a generalization, since now the statement in (a) involves a quasi-polynomial instead of a polynomial. One possibility that i see is that even though $R$ is concentrated in non-negative degrees, $M$ may now be non-zero in negative degrees as well. But there can be finitely many such negative degrees since $M$ is finitely generated and $R$ is positively graded. So i don't think that the existence of finitely many negative components of $M$ would affect the Hilbert polynomial.
Although I don't understand your question very well, let me just say that the Theorem 13.2 in Matsumura says something about the Hilbert series of a finitely generated positively graded $R$-module (this is why $f\in\mathbb Z[t]$). The corollary on the bottom of page 95 gives that the Hilbert function agrees with a polynomial only in the case that the variables have degrees $d_1=\dots=d_r = 1$.
Theorem 4.4.3 in Bruns and Herzog deals with $\mathbb Z$-graded modules and is focused on Hilbert (quasi)polynomial and Hilbert functions rather than on Hilbert series which is treated earlier in Proposition 4.4.1.
It also seems that you want to know why the Hilbert poly is in fact a quasi-polynomial in this case. This follows from 4.4.10 and it is a question related to generating functions and their representation as a rational fraction.