We study definability of second-order generalized quantifiers. We show that the question whether a second-order generalized quantifier Q₁{\mathcal{Q}}_1 is definable in terms of another quantifier Q₂{\mathcal{Q}}_2, the base logic being monadic second-order logic, reduces to the question if a quantifier Q^*₁{\mathcal{Q}}^{\star}_1 is definable in FO(Q^*₂, < ,+,×){\rm FO}({\mathcal{Q}}^{\star}_2,<,+,\times) for certain first-order quantifiers Q^*₁{\mathcal{Q}}^{\star}_1 and Q^*₂{\mathcal{Q}}^{\star}_2. We use our characterization to show new definability and non-definability results for second-order generalized quantifiers. In particular, we show that the monadic second-order majority quantifier Most¹ is not definable in second-order logic.