What do the Hurwitz quaternions have to do with the Hurwitz quaternion order?

It appears that the term does not refer to a generalization of the relationship of Hamilton's quaternions to the Hurwitz quaternions (as one might expect) but rather the term as defined there is just a specific order in a specific quaternion algebra other than $\mathbb H$. Go figure.

What they have to do is they are both related to things studied by Hurwitz. The "Hurwitz quaternions" are an object typically considered in number theory (and as a number theorist, if I hear "Hurwitz order" that's what I'll think of), and were directly studied by Hurwitz.

The other "Hurwitz order" comes up in geometry because it is related to Hurwitz surfaces, which are compact Riemann surfaces that have $84(g-1)$ automorphisms ($g =$ genus). This is the maximal number possible by a theorem of Hurwitz. The smallest possible genera for Hurwitz surfaces are 3, 7 and 14. The first two have automorphism groups of the form PSL(2,$q$), whereas the third has automorphism group the triangle group (2,3,7). The latter group is essentially the unit group in the other "Hurwitz order," hence the name, though as far as I know was not actually studied by Hurwitz.