Actually, fake elliptic curves are discussed in Chapter 9 of Lang's Introduction to algebraic and abelian functions.
In order to describe the map to $GSP(4,Q)$, recall that there is the standard antiinvolution $x\to x'$ on the quaternion $Q$-algebra $H$ such that both $tr(x)=x+x'$ and $Norm(x)=xx'=x'x$ are rational numbers for all $x\in H$. Every positive antiinvolution of $H$ is of the form $x \to \gamma^{-1}x'\gamma$ where $\gamma$ is a fixed nonzero (invertible) element of $H$ such that $\gamma^2$ is a negative rational number and therefore $\gamma'=-\gamma$. This gives rise to the alternating nondegenerate $Q$-bilinear form $$E: H \times H \to Q, x,y \mapsto tr(\gamma x y').$$ Now let us consider the following faithful action of the multiplicative group $H^G$ of $H$ on the $Q$-vector space $V=H$: $u(x)=x u'$ $u(x)=x u'$$ for all $u\in H^$ and $x \in V=H$ and $u \in G$. Then
$$E(u(x),u(y))=tr(\gamma x u'u y')=(u'u)tr(\gamma x y')$$ y'),$$
which means that $$=Norm(u) $E(u(x),u(y))=Norm(u) E(x,y).$$
This gives us the embedding $$H^* G \hookrightarrow GSp(V,E)\cong GSP(4.Q).$$ to GSP(V,E)\cong GSP(4.Q)$.
The same construction over arbitrary commutative $Q$-algebras $R$ gives us the desired embedding of the corresponding $Q$-algebraic groups.
