  We construct a fundamental system of a $q$-difference Lax pair of rank $N$ in
terms of 5d Nekrasov functions with $q=t$. Our fundamental system degenerates
by the limit $q\to 1$ to a fundamental system of a differential Lax pair, which
yields the Fuji-Suzuki-Tsuda system. We introduce tau functions of our system
as Fourier transforms of 5d Nekrasov functions. Using asymptotic expansions of
the fundamental system at $0$ and $\infty$, we obtain several determinantal
identities of the tau functions.
