［Purposes］ Consider a class of quasilinear biharmonic equations in the plane. ［Methods］ This problem can be transformed into an ordinary differential equation boundary value problem via the method of radial symmetry, and obtained its equivalent Hammerstein-type integral equation. Then, an operator equation is established in an appropriate work space, and using an inequality from the Green and the fixed point theorem of increasing operators, the main results are obtained. ［Findings］ The uniqueness and existence of positive radial solutions are obtained, and an iterative sequence for the positive solution is also offered. ［Conclusions］ Two examples are given to illustrate that the conclusion has a wide application, and the result extends and generalizes the existing study in the literature.