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We evaluate the nucleon axial form factor, $G_A(q^2)$, and inducedpseudoscalar form factor, $G_p(q^2)$, as well as the pion-nucleon form factor,$G_{\pi N N}(q^2)$, in lattice QCD. We also evaluate the corresponding nucleonto $\Delta$ transition form factors, $C_5^A(q^2)$ and $C_6^A(q^2)$, and thepion-nucleon-$\Delta$ form factor $G_{\pi N\Delta}(q^2)$. The nucleon formfactors are evaluated in the quenched theory and with two degenerate flavors ofdynamical Wilson fermions. The nucleon to $\Delta$ form factors, besides Wilsonfermions, are evaluated using domain wall valence fermions with staggered seaquark configurations for pion masses as low as about 350 MeV. Using these formfactors, together with an evaluation of the renormalized quark mass, weinvestigate the validity of the diagonal and non-diagonal Goldberger-Treimanrelations. The ratios $G_{\pi N\Delta}(q^2)/G_{\pi NN}(q^2)$ and$2C_5^A(q^2)/G_A(q^2)$ are constant as a function of the momentum transfersquared and show almost no dependence on the quark mass. We confirm equality ofthese two ratios consistent with the Goldberger-Treiman relations extracting amean value of $1.61(2)$.

Axial Nucleon and Nucleon to Delta form fractors and the Goldberger-Treiman Relations from Lattice QCD