  Linear quiver ${\cal N}=1$ 5d gauge theory in $\Omega$ background is
considered. It is shown that under certain restrictions on the VEV's of the
adjoint scalar field corresponding to the first node, only the array of Young
diagrams, such that the first diagram is a single column and the others are
empty, contribute to the partition function. Furthermore it is proved that this
partition function in a simple way is related to the expectation values of
Baxter's $Q$ operator (at specific discrete values of the spectral parameter)
in the gauge theory with the special node removed. Using known expression of
the partition function in the $U(1)$ quiver, Baxter's T-Q difference equations
are established and explicit expressions for the VEV of the $Q$ operator in
terms of generalized q-deformed Appel's functions is found. Finally the
corresponding expressions for the 4d limit are derived.
