A　book　(Formula　presented.)　is　a　graph　which　consists　of　(Formula　presented.)　triangles　sharing　a　common　edge.　In　this　paper,　we　study　Ramsey　numbers　of　quadrilateral　versus　books.　Previous　results　give　the　exact　value　of　(Formula　presented.)　for　(Formula　presented.).　We　aim　to　determine　the　exact　value　of　(Formula　presented.)　for　infinitely　many　(Formula　presented.).　To　achieve　this,　we　first　prove　that　(Formula　presented.)　for　(Formula　presented.)　and　(Formula　presented.).　This　improves　upon　a　result　by　Faudree,　Rousseau,　and　Sheehan which　states　that　(Formula　presented.)　Combining　the　new　upper　bound　and　constructions　of　(Formula　presented.)　-free　graphs,　we　are　able　to　determine　the　exact　value　of　(Formula　presented.)　for　infinitely　many　(Formula　presented.).　As　a　special　case,　we　show　(Formula　presented.)　for　all　prime　powers　(Formula　presented.).　©　2022　Wiley　Periodicals　LLC.