  The $q$-analogs of Bernoulli and Euler numbers were introduced by Carlitz.
Similar to the recent results on the Hankel determinants for the $q$-Bernoulli
numbers established by Chapoton and Zeng, we determine parallel evaluations for
the $q$-Euler numbers. It is shown that the associated Favard-type orthogonal
polynomials for $q$-Euler numbers are given by a specialization of the big
$q$-Jacobi polynomials, thereby leading to their corresponding Jacobi continued
fraction expression, which eventually serves as a key to our determinant
evaluations.
