An application of the axiom of choice? In my mind the axiom of choice is about being able to choose one element from each subset of some universal set $X$. I dont see the connection to the Axiom of choice in the part of the proof below. 
Let $u(x): \mathbb{R}^2 \rightarrow \mathbb{R}$ with $u(x,2) > u(x,1) \; \forall  \; x \in \mathbb{R}$.  By density of $\mathbb{Q}$ in $\mathbb{R}$, $ \exists \, q(x) \in \mathbb{Q} : u(x,2) > q(x) > u(x,1).$ 
Having done that for every $x \in \mathbb{R}^+$, we obtain a function $q: \mathbb{R}^+ \rightarrow \mathbb{Q}$ by the Axiom of choice.
 A: While we do make infinitely (in fact, continuum-many) choices here, we do not need the Axiom of Choice in order to show that a such function $q$ exists. This is because we can explicitly describe an enumeration of $\Bbb Q$ and then simply always pick the "first" rational in the allowed range.
A: To directly address the question 

How is the choosing of one element from each subset related to "we obtain a function $q: \mathbb{R}^+ \rightarrow \mathbb{Q}$"?

without going into whether you actually need Choice for this:
Say you for each $x\in \Bbb R^+$ have a prescribed, non-empty $U_x\subseteq \Bbb Q$ (specifically, the interval $(u(x, 1), u(x, 2))\cap \Bbb Q$ of rational numbers). Then what the Axiom of Choice says is exactly that there exists a function $q:\Bbb R^+\to \Bbb Q$ that for each $x\in \Bbb R^+$ gives an element $q(x)\in U_x$.
A: For every $x$, the set $Q_x=\{q\in\Bbb Q\mid u(x,2)>q>u(x,1)\}$ is non-empty. Therefore using the axiom of choice we can choose $q_x\in Q_x$ for all $x\in\Bbb R^+$ and thus we have a function $q(x)=q_x$.
However, the rational numbers are countable, so we don't need the axiom of choice. We can take the rational with the least numerator and the least denominator of that numerator (when considering a rational as a fraction $\frac pq$ with $p$ and $q$ integers without a common divisor other than $1$, and $q>0$).
A: We have a collection of non empty sets
$$\big((u(x, 1), u(x,2))\cap {\mathbb Q}\big)_{x\in {\mathbb R}}$$
The axiom of choice says that there is a collection of elements $(q_x)_{x\in {\mathbb R}}$ such that
$$\forall x\in \mathbb{R}, q_x\in (u(x, 1), u(x,2))\cap {\mathbb Q}$$
However as @HagenvonEitzen pointed out, this argument is not needed here.
