In this paper we first present a CG-type method for inverse eigenvalue problem of constructing real and symmetric matrices $M$, $D$ and $K$ for the quadratic pencil $Q(\lambda)=\lambda^2M+\lambda D+K$, so that $Q(\lambda)$ has a prescribed subset of eigenvalues and eigenvectors. This method can determine the solvability of the inverse eigenvalue problem automatically. We then consider the least squares model for updating a quadratic pencil $Q(\lambda)$. More precisely, we update the model coefficient matrices $M$, $C$ and $K$ so that (i) the updated model reproduces the measured data, (ii) the symmetry of the original model is preserved, and (iii) the difference between the analytical triplet $(M, D, K)$ and the updated triplet $(M_{\text{new}}, D_{\text{new}}, K_{\text{new}})$ is minimized. In this paper a computationally efficient method is provided for such model updating and numerical examples are given to illustrate the effectiveness of the proposed method.