  Let $\mu$ be a positive Borel measure on $[0,1)$. If $f \in H(\mathbb{D})$
and $\alpha>-1$, the generalized integral type Hilbert operator defined as
follows:
  $$\mathcal{I}_{\mu_{\alpha+1}}(f)(z)=\int^1_{0}
\frac{f(t)}{(1-tz)^{\alpha+1}}d\mu(t), \ \ \ z\in \mathbb{D} .$$ The operator
$\mathcal{I}_{\mu_{1}}$ has been extensively studied recently. In this paper,
we characterize the measures $\mu$ for which $\mathcal{I}_{\mu_{\alpha+1}}$ is
a bounded (resp., compact) operator acting between the Bloch space $\mathcal
{B}$ and Bergman space $ A^{p}$, or from $A^{p}(0<p<\infty)$ into $ A^{q}(q\geq
1)$. We also study the analogous problem in Bergman spaces $A^{p}(1 \leq p\leq
2)$. Finally, we determine the Hilbert-Schmidt class on $A^{2}$ for all
$\alpha>-1$.
