There is an interesting relation between the quantum periods on a certain limit of local $\mathbb{P}^1\times \mathbb{P}^1$ Calabi-Yau space and TBA (Thermodynamic Bethe Ansatz) system appeared in studies ABJM (Aharony-Bergman-Jafferis-Maldacena) theory. We propose one-parameter generalization relation. Furthermore, we derive differential operators for TBA-like equation various limits generalized