Feynman has a lecture book on gravity available. However, it is not strictly speaking a book on quantum gravity. It is a book where classical gravity is built out of quantum field theory methods : how (and whether) Einstein's gravity uniquely results from massless spin 2 exchange. The construction is plausible, yet not necessarily unique mathematically speaking.
Specifically, the last chapters of the lecture where he attempted the full quantum version (to renormalize loops) have not been included, as Feynman decided because he was not successful and did not feel the material could be useful. Many parts of the book which are available are also somehow outdated and mostly have a historical and pedagogical value. It is nonetheless interesting to read how Feynman attempts to tackle new problems.