Solutions to variant of $q$-hypergeometric equation of degree 2 associated with $q$-middle convolution

We investigate the integral representations of solutions to the variant of $q$-hypergeometric equation of degree 2 obtained through $q$-middle convolution by using transformation formulas for $q$-hypergeometric series. We show the correspondence between these integral solutions and solutions obtained by other methods. We also show the linear relationships among the integral solutions.