A Gaussian interval quadrature formula with respect to the positive weight \(w\) is a quadrature formula of the form \[ \int _a^bfw\,dx\approx \sum _{k=1}^n \frac {\mu _k}{2h_k}\int _{x_k-h_k}^{x_k+h_k}fw\,dx, \] which integrates exactly all polynomials of degree less than \(2n\). This paper is concerned with the existence and uniqueness of Gaussian interval quadrature formula for the generalized Laguerre weight function \(w(x)=x^{\alpha} e^{-x}\), \(\alpha >-1\) on \((a,b)=(0,+\infty )\). In this case, the authors denote by \({\mathbf H}_n^H\) the set of admissible lengths \({\mathbf H}_n^H=\{ {\mathbf h}\in \mathbb R ^n \mid h_i\geq 0,\sum _ {k=1}^nh_k H\} \) and by \({\mathbf X}_n({\mathbf h})=\{ {\mathbf x}\in \mathbb R ^n \mid 0