I take that your primary goal is to know what Feynman thought of Hawking's work. While it is possible that they have met I would consider it unlikely given that Feynman mentioned several times how his attendance of the Chapel Hill conference (on general relativity) made such a bad impression on him that he never attended another conference on that theme, so they largely went around different social circles.
Yet, regarding Hawking radiation it is clear that he never understood it completely. If you search for "Feynman's last blackboard" you'll find the photograph of the blackboard in Feynman's Caltech Office at the time of his death. In the right upper corner there is a list of things "to learn", they are: Bethe Ansatz Probl., Kondo (dashed, so presumably he got that one), 2D Hall, accel. temp. and non-linear classical hydro.[If you have trouble finding it using your-favorite-search-engine look for the Reviews of Modern Physics about the Unruh Effect, authors are Matsas, Crispino and Higuchi]
Accel. Temp. is a clear reference to Acceleration Temperature, that is the Unruh Effect. While Hawking Radiation and Unruh Effect are different physical phenomena (that is, they have different causes), they are very closely related, both physically and in their mathematical derivation, therefore it seems unreasonable to think that he understood Hawking's work, and other related to quantum gravitation, since he did not understand the Unruh Effect, arguably the simplest example of related phenomena.
The other answer mentions stimulated emission by rotating black holes (sometimes people call it Zel'dovich Effect in the literature). By Hawkings own recollections, he first thought of Hawking radiation while visiting the USSR in the 1970s, and there Zel'dovich showed him that a electrically charged black hole would amplify electromagnetic waves. He (Hawking) then made the connection with stimulated and spontaneous emission, and back in the UK finished the calculations. [We'll, it's embarrassing but I cannot remember exactly where he wrote that, I'm not in my office for some time now. This version of the story is either in the book "300 years of gravitation" (Hawking was one of it's editors) or in Kip Thorne's "Black Holes and Time Warps". Sorry for giving no concrete reference, I would bet on Thorne's book right now, also because if Feynman showed the math to Kip Thorne's students then surely it will be mentioned in Thorne's recollection on the history of GR.]
Hope it helps to dig further references, but this is the extend of which I'm aware.