Blowup relations on $\mathbb{C}^2/\mathbb{Z}_2$ from Nakajima-Yoshioka blowup relations

We obtain bilinear relations on Nekrasov partition functions, arising from study of tau functions of quantum $q$-Painlev\'e equations, from Nakajima-Yoshioka blowup relations by an elementary algebraic approach. Additionaly, using this approach, we prove certain relations on Nekrasov partition functions, modified by Chern-Simons term.