Following a similar technique than the one in Pandoc: Markdown to PDF, without cutting off code block lines that are too long you can write a file named, for example quote-setup.tex
containing:
\usepackage{framed}
\let\oldquote=\quote
\let\endoldquote=\endquote
\renewenvironment{quote}{\begin{framed}\begin{oldquote}}{\end{oldquote}\end{framed}}
and compile with:
pandoc example.md -H quote-setup.tex -o example.pdf
IMHO, the frame is "too agressive" visually. I would prefer a more subtle effect such as a shaded background. This can be achieved with:
\usepackage{framed}
\usepackage{xcolor}
\let\oldquote=\quote
\let\endoldquote=\endquote
\colorlet{shadecolor}{orange!15}
\renewenvironment{quote}{\begin{shaded*}\begin{oldquote}}{\end{oldquote}\end{shaded*}}
So, for example:
# Example section
Now, a quote:
> Proin iaculis tellus sed fringilla elementum. Suspendisse porta tincidunt magna.
> Mauris cursus pharetra lacinia. Mauris ac auctor nibh. Duis laoreet scelerisque
> leo, ac pellentesque lorem feugiat tempus. In eu rhoncus velit. Ut bibendum nibh
> ut feugiat semper.
will produce: