The effect of nucleon coupling constants on the moment of inertia of proto-neutron star (PNS) PSR J0740+6620 is examined with relativistic mean-field theory in consideration of a baryon octet. Here, nucleon coupling parameters DD-MEI, NL1, NL2, TW99, and GM1 are used. Under the constraints of the observed mass M = 2.05-2.24 M(circle dot )of PNS PSR J0740+6620 (taking the observed mass of neutron star (NS) PSR J0740+6620), the radius of the PNS calculated by us with GM1 is the smallest, R = 14.63-13.44 km, while the radius calculated with TW99 is the largest, R = 17.46-17.07 km. The radii calculated from the other three sets of nucleon coupling parameters are between the two values mentioned above. The maximum value of the moment of inertia calculated with the nucleon coupling parameters TW99, NL1, DD-MEI, NL2, and GM1 decreases in turn. Under the constraint of the observed mass of PNS PSR J0740+6620 (taking the observed mass of NS PSR J0740+6620), the moment of inertia calculated with NL1, NL2, DD-MEI, and TW99 increases with the increase of the central energy density and mass, and decreases with the increase of the radius. The moment of inertia calculated with GM1 decreases with the increase of central energy density and mass, and increases with the increase of the radius. The moment of inertia of the PNS PSR J0740+6620 calculated by us with five groups of nucleon coupling parameters is between 2.465 x 10(45)-2.050 x 10(45) g cm(2) (GM1) and 3.597 x 10(45)-3.883 x 10(45) g cm(2 )(TW99).