Summary: We show that TBA equations defined by the BPS spectrum of \(5d\) \(\mathcal{N} =1\) \(SU(2)\) Yang-Mills on \(S^1\times \mathbb{R}^4\) encode the \(q\)-Painlevé \(\mathrm{III}_3\) equation. We find a fine-tuned stratum in the physical moduli space of the theory where solutions to TBA equations can be obtained exactly, and verify that they agree with the algebraic solutions to \(q\)-Painlevé. Switching from the physical moduli space to that of stability conditions, we identify two one-parameter deformations of the fine-tuned stratum, where the general solution of the q-Painlevé equation in terms of dual instanton partition functions continues to provide explicit TBA solutions. Motivated by these observations, we propose a further extensions of the range of validity of this correspondence, under a suitable identification of moduli. As further checks of our proposal, we study the behavior of exact WKB quantum periods for the quantum curve of local \(\mathbb{P}^1\times\mathbb{P}^1\).