Using data samples with an integrated luminosity of 5.85 fb(-1) collected at center-of-mass energies from 4.61 to 4.95 GeV with the BESIII detector operating at the BEPCII storage ring, we measure the cross section for the process e(+)e(-) -> K+ K- J/psi. A new resonance with a mass of M = 4708(-15)(+17) +/- 21 MeV/c(2) and a width of Gamma = 126(-23)(+27) +/-+/- 30 MeV is observed in the energy-dependent line shape of the e(+)e(-) -> K+ K- J/psi. cross section with a significance over 5 sigma. The K(+)J/psi system is also investigated to search for charged charmoniumlike states, but no significant Z(cs)(+) states are observed. Upper limits on the Born cross sections for e+e- -> K-Z(cs)(3985)(+)/K-Z(cs)(4000)(+) + c:c: with Z(cs)(3985)(+/-)/Z(cs)(4000)(+/-) -> K(+/-)J/psi are reported at 90% confidence levels. The ratio of branching fractions {[B(Z(cs)(3985)(+) -> K+J/psi)]/B[Z(cs)(3985)(+) -> ((D) over bar D-0(s)(+) + (D) over barD-0(s)+)]} is measured to be less than 0.03 at 90% confidence level.