Inspired by a question of Kra, Moreira, Richter, and Robertson, we prove two new results about infinite polynomial configurations in large subsets of the rational numbers. First, given a finite coloring of $\mathbb{Q}$, we show that there exists an infinite set $B = \{b_n : n \in \mathbb{N}\} \subseteq \mathbb{Q}$ such that $$\{b_i, b_i^2 + b_j : i < j\}$$ is monochromatic. Second, we prove that every subset of positive density in the rational numbers contains a translate of such an infinite configuration. The corresponding results in the integers are both known to be false, so our results provide natural and relatively simple examples of combinatorial structures that distinguish between the Ramsey-theoretic properties of the rational numbers and the integers. The proofs of our main results build upon methods developed in a series of papers by Kra, Moreira, Richter, and Robertson to translate from combinatorics into dynamics, where the core of the argument reduces to understanding the behavior of certain polynomial ergodic averages. The new dynamical tools required for this analysis are a Wiener--Wintner theorem for polynomially-twisted ergodic averages in $\mathbb{Q}$-systems and a structure theorem for Abramov $\mathbb{Q}$-systems.