  An integral presentation for the scalar products of nested Bethe vectors for
the quantum integrable models associated with the quantum affine algebra
$U_q(\hat{\mathfrak{gl}}_3)$ is given. This result is obtained in the framework
of the universal Bethe ansatz, using presentation of the universal Bethe
vectors in terms of the total currents of a "new" realization of the quantum
affine algebra $U_q(\hat{\mathfrak{gl}}_3)$.
