Let $p$ be a prime. Suppose you have an Abelian scheme $A$ over $Spec\ \mathbb{Z}_p$. How do you prove that if $q$ is another prime, then the $q$torsion of $A$ injects into the torsion of $A_p$, under the reduction map?

Let me try again at an alternate answer. If $A_{\mathbb{Z}_p}$ is an abelian scheme and $\ell \neq p$ is a prime, then for any positive integer $n$, the isogeny $[\ell^n]: A \rightarrow A$ is an etale map. [If I am not mistaken, the proof of this does not require formal groups!] Since the special fiber has $\ell^{2n}$ points over the algebraic closure, by Hensel's Lemma all of the $\ell^n$torsion on $A$ is defined over the maximal unramified extension, and it follows that the reduction map over the maximal unramified extension is an isomorphism on the $\ell^n$torsion, hence an injection over $\mathbb{Q}_p$. 


If A is an abelian variety over a local field of char zero with good reduction, then any point in the kernel of reduction is in the socalled formal group. Now, to answer your question (properly understood), you want to show that the formal group does not have $q$torsion for $q \ne p$. This follows because multiplication by $q$ is invertible as a (vector of) formal power series, so is a bijection in the formal group. 


To complete Pete and Milne's answers when A is not an abelian scheme (for example, when it is the Néron model of an abelian variety over ${\mathbb Q}_p$ with not necessary good reduction), then for any $n$ prime to $p$, the kernel $A[n]$ is still étale over ${\mathbb Z}_p$ (because the tangent map at $0$ of the multiplication by $n$ is just multiplication by $n$ for any commutative algebraic group), but not necessarily finite. There is a biggest closed subscheme $H$ of $A[n]$ which is étale and finite over ${\mathbb Z}_p$. The reduction map on $H$ is injective (see Pete's proof). The generic fiber of $H$ corresponds to the points of the generic fiber of $A[n]$ having specialization mod $p$. You may read Bosch, Lütkebohmert and Raynaud ''Néron Models'', § 7.3. 

This is Hartshorne, Exercise IV.4.19: Let $\mathcal{A}/\mathbb{Z} \setminus S =: T$ be an Abelian scheme. The multiplication by $n$ morphism is flat [I don't know how to show this, but I think it can be found in KatzMazur.], so the $n$Torsion $\mathcal{A}[n] \to \mathcal{A}$ is also flat as it is a base change and $\mathcal{A} \to T$ also since it is flat. It is also proper and quasifinite, and therefore finite. So we have a finite flat group scheme. Because of $(n,p) = 1$ $\mathcal{A}[n]$ is étale over $\mathbb{Z}_{(p)}$ (how to show this?). We have for $X/S$ finite étale Consider the reduction map $\mathcal{A}[n]_\eta(\mathbf{Q}) = \mathcal{A}[n](T) \to \mathcal{A}[n]_p(\mathbf{F}_p)$ for $(n,p) = 1$, confer Liu, Chapter 10.1.3. Liu, Proposition 10.1.40(b) gives us onepoint fibres of the reduction map. 

