In this paper, the authors study the {\it parameterised quadratic inverse eigenvalue problem} (PQIEP) with given distinct eigenvalues. In this case, the PQIEP is reduced to an equivalent multiparameter eigenvalue problem. Then, a sufficient solvability condition for the PQIEP is presented. \par To find the approximate solution of the PQIEP, the authors employ the Newton method based on the smooth $QR$ decomposition with column pivoting and prove its locally quadratic convergence. Finally, some numerical examples are given to show the effectiveness of the method.