We report the first measurement of the full angular distribution for inclusive J/psi -> mu(+)mu(-) decays in p + p collisions at root s = 510 GeV. The measurements are made for J/psi transverse momentum 2 < p(T) < 10 GeV /c and rapidity 1.2 < y < 2.2 in the Helicity, Collins-Soper, and Gottfried-Jackson reference frames. In all frames the polar coefficient lambda theta is strongly negative at low p(T) and becomes close to zero at high p(T), while the azimuthal coefficient lambda phi is close to zero at low p(T), and becomes slightly negative at higher p(T). The frame-independent coefficient lambda is strongly negative at all p(T) in all frames. The data are compared to the theoretical predictions provided by nonrelativistic quantum chromodynamics models.