Polynomials on vector lattices possess interesting order properties, and classes of polynomials on Banach lattices, defined in mixed terms of norm and order, have a rich structure. This is why the subject draw growing attention of researchers. The classes of (p,q)-convex and (p,q)-concave linear operators in Banach lattices, as well as the conceptions of type and cotype play an important role in the theory of Banach lattices and bounded linear operators. All these concepts and many related results may be naturally transplanted to the environment of quasi-Banach spaces. Convexity arguments do not work well in arbitrary quasi-Banach spaces and this led to developing of new approaches and techniques. The aim of this work is to extend the above circle of ideas from linear case to the polynomial setting and examine convexity conditions for homogeneous polynomials on quasi-Banach lattices.