We study a model of quantum field theory with ``Yukawa-like'' interaction λ∫Φb+(x)Φb−(x)Φa(x)d x between nucleons (b) and mesons (a). It is a version of Nelson's model with relativistic kinematics and has been renormalized by J. P. Eckmann. The infinite mass renormalization is a power series in λ2, chosen in such a way as to confer on the renormalized Hamiltonian Ĥ the correct relativistic single particle spectrum. Physical one nucleon states are given by a modified Friedrichs one-particle expansion constructed by Eckmann. The Heisenberg picture's creation-annihilation operator for dressed nucleons and mesons are studied in detail, as a preparation for the construction of the correspondent asymptotic fields, carried through, in this paper, for the mesons fields in general and for the nucleon fields on particular states (the general case is treated in the second paper of this series). Analytic properties of the interacting fields in λ are proved and commutation relations of the asymptotic fields are established. Moreover, strong asymptotic states are constructed as well as isometric wave operators. Finally some reduction formulas for the meson-nucleon scattering are derived.