The Hurwitz-type Euler zeta function $\J(z,q)$ is defined by the series $$\J(z,q)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+q)^{z}},$$ for Re$(z)>0$ and $q\neq0,-1,-2,\ldots,$ and it can be analytic continued to the whole complex plane. An asymptotic expansion for $\zeta_{E}'(-m,q)$ has been proved based on the calculation of Hermite's integral representation for $\J(z,q).$