A search is performed for exclusive high-mass gamma gamma -> WW and gamma gamma -> ZZ production in proton-proton collisions using intact forward protons reconstructed in near-beam detectors, with both weak bosons decaying into boosted and merged jets. The analysis is based on a sample of proton-proton collisions collected by the CMS and TOTEM experiments at root s = 13 TeV, corresponding to an integrated luminosity of 100 fb(-1). No excess above the standard model background prediction is observed, and upper limits are set on the pp -> pWWp and pp -> pZZp cross sections in a fiducial region defined by the diboson invariant mass m(VV) > 1 TeV (with V = W, Z) and proton fractional momentum loss 0.04 < xi < 0.20. The results are interpreted as new limits on dimension-6 and dimension-8 anomalous quartic gauge couplings.