Is there a relationship between the cross product and quaternion multiplication?

Yes. If we let a vector be written as the coefficients of a quaternion with zero "real" component, i.e. $[a,\ b,\ c] = ai+bj+ck$, then the cross product is simply the quaternion product with the real part omitted. See also: http://en.wikipedia.org/wiki/Cross_product#Quaternions

You may have heard along the way that vector cross products only exist in 3 and 7 dimensions. Why 3 and 7? Because we can "mimic" the cross product in $\mathbb{R}^3$ with quaternions, and likewise we can use octonions to mimic the cross product in $\mathbb{R}^7$.

Why not any other dimension? As it turns out, octonions represent the highest-dimension normed division algebra. So we can go no higher!

See the product formula at the end of this section about quaternions. Note that (following the same notation) in particular for $v,w\in \mathbb{R}^3$ $$(0,v)(0,w) = (-v\cdot w, v\times w).$$