A possible simple approach would be to use a minipage inside the quote environment:
\documentclass{article}
\usepackage{lipsum}
\begin{document}
\lipsum[1]
\begin{quote}
\begin{minipage}{\linewidth}
\stepcounter{footnote}
\renewcommand\thempfootnote{\arabic{footnote}}
\lipsum[1] text\footnote{this is a footnote} text text text.
\end{minipage}
\end{quote}
\end{document}

Here's an enhanced version of my answer: now everything is dome through a new environment fnquote that redefines and steps appropriately the counter and formats the footnote mark as in the image (suppressing the indentation):
\documentclass{article}
\usepackage{lipsum}
\makeatletter
\newenvironment{fnquote}
{\begin{quote}
\begin{minipage}{\linewidth}\stepcounter{footnote}
\renewcommand\thempfootnote{\arabic{footnote}}
\renewcommand\@makefntext[1]{\noindent\makebox[1.2em][l]{\@makefnmark}##1}}
{\end{minipage}\end{quote}}
\makeatother
\begin{document}
\lipsum[1]
\begin{fnquote}
\lipsum[1] text\footnote{this is a footnote} text text text.
\end{fnquote}
\end{document}

A new version; a new command \qtfootnote is available allowing to manually specify the value for the footnote marker:
\documentclass{article}
\usepackage{lipsum}
\newcounter{tmpfn}
\makeatletter
\newenvironment{fnquote}
{\begin{quote}
\begin{minipage}{\linewidth}
\renewcommand\thempfootnote{\arabic{mpfootnote}}
\renewcommand\@makefntext[1]{\noindent\makebox[1.2em][l]{\@makefnmark}##1}
}
{\end{minipage}\end{quote}}
\makeatother
\newcommand\qtfootnote[2]{%
\setcounter{tmpfn}{\thempfootnote}%
\setcounter{mpfootnote}{#1}\addtocounter{mpfootnote}{-1}%
\footnote{#2}%
\setcounter{mpfootnote}{\thetmpfn}}
\begin{document}
\lipsum[1]
\begin{fnquote}
\lipsum[1] text\qtfootnote{5}{this is a footnote} text text text.
\end{fnquote}
\end{document}

And the fourth version, now the code has been greatly simplified by using the features provided by the nccfoots package; now the footnote marker can be freely assigned (arabic, alphabetic, roman, etc.) in an easy way:
\documentclass{article}
\usepackage{nccfoots}
\newcommand\Text{Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit,
vestibulum ut, placerat ac, adipiscing vitae, felis. Curabitur dictum gravida
mauris. Nam arcu libero, nonummy eget, consectetuer id, vulputate a, magna.
Donec vehicula augue eu neque.}
\makeatletter
\newenvironment{fnquote}
{\begin{quote}\begin{minipage}{\linewidth}
\renewcommand\@makefntext[1]{\noindent\makebox[1em][l]{\@makefnmark}##1}}
{\end{minipage}\end{quote}}
\makeatother
\newcommand\qtfootnote[2]{\Footnotemark{#1}\Footnotetext{#1}{#2}}
\begin{document}
\Text
\begin{fnquote}
\Text text text text text\qtfootnote{4}{this is a footnote} text text.
\end{fnquote}
\begin{fnquote}
\Text text text text text\qtfootnote{d}{this is a footnote} text text.
\end{fnquote}
\begin{fnquote}
\Text text text text text\qtfootnote{IV}{this is a footnote} text text.
\end{fnquote}
\end{document}

parnotesis worth reading the documentation. It resulted BTW from this TeX.SX question: http://tex.stackexchange.com/questions/34746/placing-footnotes-or-endnotes-at-the-end-of-the-current-paragraph. – Speravir Mar 10 '12 at 02:51parnotesis much easier than hacking the footnote code. – Chel Mar 10 '12 at 15:43\theparnoteor similar? See my comment to @drozzy, too. – Speravir Mar 10 '12 at 22:48\makeatletter\def\parnotesetto#1{\c@PN@t=#1\c@PN@n=#1}\makeatothershould work. I'll work on also adding a\parnote[...]{...}command withnccfoots-like behavior. – Chel Mar 11 '12 at 04:50\begin{quote}\parnotesetto{4}French text...\parnotes\parnotessetto{4}Tenslation...\parnotes\end{quote}. – Chel Mar 11 '12 at 05:06