A search for exclusive or quasi-exclusive gamma gamma -> W (+) W (-) production, via pp -> p (({*})) W (+) W (-) p (({*})) -> p (({*})) mu (+/-)e(a{''}) p (({*})) at TeV, is reported using data corresponding to an integrated luminosity of 19.7 fb(-1). Events are selected by requiring the presence of an electron-muon pair with large transverse momentum p (T)(mu (+/-)e(a{''})) > 30 GeV, and no associated charged particles detected from the same vertex. The 8 TeV results are combined with the previous 7 TeV results (obtained for 5.05 fb(-1) of data). In the signal region, 13 (2) events are observed over an expected background of 3.9 +/- 0.6 (0.84 +/- 0.15) events for 8 (7) TeV, resulting in a combined excess of 3.4 sigma over the background-only hypothesis. The observed yields and kinematic distributions are compatible with the standard model prediction for exclusive and quasi-exclusive gamma gamma -> W (+) W (-) production. Upper limits on the anomalous quartic gauge coupling operators a (0,C) (W) (dimension-6) and f (M0,1,2,3) (dimension-8), the most stringent to date, are derived from the measured dilepton transverse momentum spectrum.