Let \(X_n (a_1, \dots, a_k)\) be the \(n\)-dimensional smooth projective toric variety of Picard number 2 which was defined by P. Kleinschmidt [Äquationes Math. 35, No. 2/3, 254-266 (1988; Zbl 0664.14018)] as a generalization of Hirzebruch surfaces. It is shown that \(X_n (a_1, \dots, a_k)\) has a projective embedding \(\varphi : X_n (a_1, \dots, a_k) \to \mathbb{P}^r\) such that \(\varphi (X_n (a_1, \ldots, a_k))\) is an intersection of quadratic hypersurfaces.