In order to extend the basic theory of boundary value problems for nonlinear quantum difference equations,the existence of solutions for a class of second order three-point nonlinear q-differential equations with a first order q-differential on a nonlinear interval is studied. Firstly, changing the order of integration formula of double q-integral with infinite limit generalized integral is given and proved. Secondly, the Green function of the boundary value problem of second-order three-point linear q-difference equation on the infinite interval is calculated and the property of Green function is studied. Next, the integral operator T is constructed on the abstract space, and the Leray-Schauder continuous theorem is used to obtain the existence of the solution of the boundary value problems for the second-order three-point nonlinear q-difference equation on the infinite interval. Finally, an example is given to illustrate the validity of the results. The research results have important significance for the development of quantum calculus and its application in the fields of mathematical physics.