In general, I cannot say very much. Let $K = \mathbb Q(\zeta_d)$. Suppose $V_{\mathbb Q}$ is a simple $\mathbb QG$-module corresponding to a simple factor $M_n(K)$. Then $V_{\mathbb Q}$ comes from restriction of scalars along $K/\mathbb Q$ of a $KG$-module $V$ which is absolutely irreducible. Then $V$ is faithful if and only if $V_\mathbb Q$ is faithful. So it suffices to understand when $KG$ has a faithful absolutely irreducible module. This decomposes into two parts:

1. When does $G$ have a faithful irreducible representation with character valued in $K$?

2. When is a character from 1. defined over $K$?

The question of when a group has a complex faithful irreducible representation is discussed in the following MO thread: https://mathoverflow.net/questions/57129/which-finite-groups-have-faithful-complex-irreducible-representations. 
The second point is a question of Schur indices and I cannot say much about it in general.

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In the case of a (finite) $p$-group, then there are neat criteria: a $p$-group $P$ has a faithful complex irreducible representation if and only if $Z(P)$ is cyclic [1]. Furthermore Schur indices vanish for $p$-groups when $p > 2$. The second point was used by Ford to prove the following theorem:

> **Theorem [2]:** Each (nontrivial) irreducible rational representation of a finite $p$-group is induced from the irreducible faithful rational representation of degree $p-1$ on a section of order $p$.

That is, each simple factor of $\mathbb QP$ for $p > 2$ is of the form $M_n(\mathbb Q(\zeta_p))$.

> **Corollary:** For a $p$-group $P$ with $p > 2$, there is a simple factor $M_n(\mathbb Q (\zeta_d))$ of $\mathbb QP$ corresponding to a faithful representation if and only if $P$ has cyclic center and $d = p$.

So, the Heisenberg example is a general phenomenon for odd $p$-groups.

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*References:*

[1] Isaacs, *Character Theory*, Theorem 2.32

[2] Ford, Charles E. "Characters of p-groups." In Proc. Am. Math. Soc, vol. 101, no. 4, pp. 595-600. 1987.