It's worth pointing out that in the context of Borel chromatic numbers, there is a canonical reason for a graph to not have countable chromatic number. Kechris, Solecki, and Todorcevic showed that there is a graph $\mathcal G_0$ such that a Borel graph $\mathcal G$ has $\chi_B(\mathcal G)\leq \aleph_0$ iff there is no continuous (graph) homomorphism from $\mathcal G_0$ to $\mathcal G$.

Proving $\mathcal G_0$ has uncountable Borel chromatic number normally goes via a simple category argument such as in Guest's answer, but as Joel pointed out, this can be seen through a forcing lens if you like. Proving the dichotomy is normally done using the Gandy-Harrington topology on $2^\mathbb{N}$ (part of an area known as effective descriptive set theory), which doesn't quite admit category arguments. However, the topology is what's known as strong Choquet (a property based on a certain infinite game), which allows one to carry out something very similar to category arguments. (It assures you that certain intersections of nonempty open sets are nonempty.) And of course one can talk instead about the Gandy-Harrington forcing. More recently, Ben Miller wrote a proof of the $\mathcal G_0$-dichotomy which doesn't use effective descriptive set theory or forcing.

As for $\mathcal G_0$ itself, here's a definition. Fix a sequence $(t_n)$ of binary sequences such that $|t_n|=n$ and the sequence is dense, meaning for any binary sequence $t$ there is some $n$ such that $t_n$ extends $t$. (In other words, the basic open sets of $2^\mathbb{N}$ defined by the $t_n$ intersect every open set of $2^\mathbb{N}$.) Then define $\mathcal G_0$ on $2^\mathbb{N}$ by

$$ x \mathcal G_0 y \Leftrightarrow \exists n [x\restriction n = y\restriction n = t_n \text{ and } x(n)=1-y(n) \text{ and } \forall m>n (x(m)=y(m)) ]$$

I'll note that this is clearly very similar to the graph $G$ in the question, and leave it as an exercise to find a homomorphism from $\mathcal G_0$ to $G$. I'll also note that while the definition of $\mathcal G_0$ does depend on your choice of $(t_n)$, the KST dichotomy (actually, a slight strengthening of it also due to KST) assures us that any two versions are homeomorphic to subgraphs of each other.

I highly recommend reading the Kechris-Solecki-Todorcevic paper if you have any interest in this topic. Not only is it the first systematic look at Borel chromatic numbers, it is very readable. If you'd prefer something more up-to-date, Kechris and Marks have a survey on Borel combinatorics coming soon. The preprint is available on either of their websites.