The principal aim of the present work is to evaluate a new class of integrals involving the product of generalized Laguerre polynomials \(L_{n}^{(\alpha )}\) and Bessel functions \(J_{\nu }\) which deduce some new results in terms of Kummer confluent hypergeometric function. Further, we discuss some special cases of our main results and deduce some well-known results. Some particular cases of these integrals involving Hermite and Laguerre polynomials are also derived. Finally, the results obtained are applied in the problem of propagation of a Laguerre–Bessel–Gaussian beam through a spiral phase plate.