A search for exclusive or quasi-exclusive γγ → W$^{+}$ W$^{−}$ production, via pp → p$^{()}$ W$^{+}$ W$^{−}$ p$^{()}$ → p$^{()}$ μ$^{±}$e$^{∓}$ p$^{()}$ at $ \sqrt{s}=8 $ 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${T}$(μ$^{±}$e$^{∓}$) > 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 3.4σ over the background-only hypothesis. The observed yields and kinematic distributions are compatible with the standard model prediction for exclusive and quasi-exclusive γγ → 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.