It is known that the Euler polynomials \(E_n(x)\) defined by the generating function \[ 2e^{tx}(e^t+1)^{-1}=\sum_{n=0}^\infty E_n(x)\frac{t^n}{n!} \] can be expressed via the Genocchi numbers corresponding to the generating function\break \(2t(e^t+1)^{-1}\). The authors find a \(q\)-analog of this relation. The resulting \(q\)-Euler numbers are different from those introduced by \textit{L. Carlitz} [Trans. Am. Math. Soc. 76, 332-350 (1954; Zbl 0058.01204)].