How would you write the set of rationals $\mathbb{Q}$ including the number $\infty$? i.e. so it is closed under division? The set would correspond to points on a number circle.
i.e. given $(a,b)\in \mathbb{Z}^2$ then the pair $(a,b)$ always corresponds to a rational number $a/b$, likewise so would $(2a,2b)$ and $(-a,-b)$.
So that if two rational numbers $(a,b)$ and $(c,d)$ are equal then $ad=bc$. (This would hold only if $\infty=-\infty$ in this set.)
Is it written as $\mathbb{Q} \cup \infty$ ?
The rationals would seem to naturally include the point $\infty$ from the pairs $(1,0)$, $(2,0)$ etc. unless one deliberately excluded it. Whereas the set of integers doesn't include infinity.
Is there a special name for the "rational infinity?"
You can call it the projective line $\mathbb{P}^1(\mathbb{Q})$ over $\mathbb{Q}$, as stated in the comments. It famously appears in the theory of modular forms as the set of "cusps"; see, for example, this math.SE question.
It is not quite closed under division, but it is closed under the action of the modular group
$$PSL_2(\mathbb{Z}) = \left\{ z \mapsto \frac{az + b}{cz + d} : a, b, c, d \in \mathbb{Z}, ad - bc = 1 \right\}$$
(which acts transitively), which is related to its appearance in the theory of modular forms. Of course it is also closed under the action of the slightly larger group $PGL_2(\mathbb{Q})$.