Torsion of an abelian variety under reduction.

Let me try again at an alternate answer. If $A_{\mathbb{Z}_p}$ is an abelian scheme of dimension $g$ 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 $\ell^{2gn}$ 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 so-called 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.