In order to enrich the basic theory of boundary value problems for fractional q-difference equations, the boundary value problems for a class of nonlinear fractional q-difference equations with p-Laplacian operator on infinite intervals were explored. Firstly, the Green function of the boundary value problem of linear fractional q-difference equation was calculated and its properties were studied. Secondly, the compactness criterion on infinite intervals was introduced and the integral operator on an abstract space was constructed. Thirdly, by selecting the initial value functions and using the monotone iterative technique, the existence of positive solutions for the boundary value problem were obtained. Finally, the validity of obtained results was verified through an example. The results show that when certain increasing condition is given to the nonlinear term f, the maximum and minimum positive solutions of the fractional q-difference equation can be obtained by establishing iterative sequences. The research results extend the existing relevant conclusions and provide theoretical reference for the further application of fractional q-difference equations in mathematics, physics and other fields.