With the standard settings (no bibliography package added); note that \bibepigraph must go before starting the bibliography.
\documentclass[a4paper,oneside]{book} % oneside just for the example
\usepackage{epigraph} % for \epigraph
\usepackage{etoolbox} % for \patchcmd
\usepackage{lipsum} % for mock text
\patchcmd{\thebibliography}{\list}{\printbibepigraph\list}{}{}
\newcommand{\bibepigraph}[2]{%
\def\printbibepigraph{\epigraph{#1}{#2}}%
}
\def\printbibepigraph{} % initialize
\begin{document}
\chapter{Title}
\epigraph{%
F.: What a filthy job!\\
I.: Could be worse.\\
F.: How?\\
I.: Could be raining.
}{Dr.~F.~Frankenstein and Igor}
\lipsum[1-2]
\bibepigraph{If anything can go wrong, it will}{Murphy}
\begin{thebibliography}{9}
\bibitem{cantor}
Georg Cantor, \textit{\"Uber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen},
Journal f\"ur die Reine und Angewandte Mathematik \textbf{77} (1874), 258--262.
\end{thebibliography}
\end{document}

If BibTeX is used, it will be like
\bibepigraph{If anything can go wrong, it will}{Murphy}
\bibliographystyle{<style>}
\bibliography{<filename>}