Ability to produce hydrogen gas on reaction with Na/K
First of all, this is not an acid-base reaction. Alkali metals are not bases, rather strong reducing agents. For example, in this site, you may find a excellent explanation. Also remember, metallic $\ce{Li}$, $\ce{Na}$, or $\ce{K}$   does not abstract acidic hydrogen from organic molecules, not necessarily. For example,  metallic $\ce{Na}$ reacts with acetylene ($\mathrm {p}K_a$ of terminal alkyne-$\ce{H}$ is $25$) to give sodium acetylide, but it does not undergo aldol condensation with acetone with more acidic $\ce{H}$, $\mathrm {p}K_a$ of which is $19.3$ (https://owl.oit.umass.edu/departments/OrganicChemistry/appendix/pKaTable.html). The reaction of acetone and metallic $\ce{Na}$ rather undergoes dimerization (e.g., see here). Another example is the reaction of triphenylmethane (the $\mathrm {p}K_a$ of $\ce{H}$ on the central carbon is $33$) and metallic $\ce{K}$ in 1,2-dimethoxyethane, which has given solutions of triphenylmethyl potassium containing various cleavage and reduction products derived from triphenylmethane (Ref.1), but authors stated that no hydrogen was evolved in the course of the preparation.

However, the given compound has potential to undergo redox reaction with alkaline metal and release $\ce{H2}$ gas as a byproduct because the resultant dianion stabilized by resonance and show hint of aromaticity (Ref.2). Tentative mechanism is illustrated in following scheme:



References:


R. O. House, V. Kramar, “The Chemistry of Carbanions. I. The Reaction of Triphenylmethane with Potassium$^1$,” J. Org. Chem. 1962, 27(12), 4146-4149 (DOI: 10.1021/jo01059a007).
G. Boche, H. Etzrodt, M. Marsch, W. Thiel, “The Dianion 1,2,3,4‐Tetraphenylcyclobutadienediide,” Angew. Chem. Int. Ed. Engl. 1982, 21(2), 133-133 (https://doi.org/10.1002/anie.198201331).