We present results from a search for strong H₂ absorption systems proximate to quasars (z_abs ≈ z_em) in the Sloan Digital Sky Survey (SDSS) Data Release 14. The search is based on the Lyman-Werner band signature of damped H₂ absorption lines without any prior on the associated metal or neutral hydrogen content. This has resulted in the detection of 81 systems with N(H₂) ∼ 10¹⁹ − 10²⁰ cm⁻² located within a few thousand km s⁻¹ from the quasar. Compared to a control sample of intervening systems, this implies an excess of proximate H₂ systems by about a factor of 4 to 5. The incidence of H₂ systems increases steeply with decreasing relative velocity, reaching an order of magnitude higher than expected from intervening statistics at Δv < 1000 km s⁻¹. The most striking feature of the proximate systems compared to the intervening ones is the presence of Ly − α emission in the core of the associated damped H I absorption line in about half of the sample. This puts constraints on the relative projected sizes of the absorbing clouds to those of the quasar line emitting regions. Using the SDSS spectra, we estimate the H I, metal and dust content of the systems, which are found to have typical metallicities of one tenth Solar, albeit with a large spread among individual systems. We observe trends between the fraction of leaking Ly − α emission and the relative absorber-quasar velocity as well as with the excitation of several metal species, similar to what has been seen in metal-selected proximate DLAs. With the help of theoretical H I-H₂ transition relations, we show that the presence of H₂ helps to break the degeneracy between density and strength of the UV field as main sources of excitation and hence provides unique constraints on the possible origin and location of the absorbing clouds. We suggest that most of these systems originate from galaxies in the quasar group, although a small fraction of them could be located in the quasar host as well. We conclude that follow-up observations are still required to investigate the chemical and physical conditions in individual clouds and to assess the importance of AGN feedback for the formation and survival of H₂ clouds.