This is the original definition of the quote
environment from article
:
\newenvironment{quote}
{\list{}{\rightmargin\leftmargin}%
\item\relax}
{\endlist}
Here is a redefined one using your requirements in the form of an MWE:

\documentclass{article}
\usepackage{lipsum}% http://ctan.org/pkg/lipsum
\renewenvironment{quote}
{\list{}{\rightmargin\leftmargin}%
\item\relax\tiny\textquotedblleft\ignorespaces}
{\unskip\unskip\textquotedblright\endlist}
\begin{document}
\lipsum[1]
\begin{quote}
\lipsum*[2]
\end{quote}
\lipsum[3]
\end{document}
The correction of \ignorespaces
at \begin{quote}
is to remove any spaces between at the start, while \unskip\unskip
does the same at \end{quote}
. Also, don't wrap the contents in a group {
}
, which provide the unwanted spacing.
Note that this will replace the existing quote
environment. If you're planning on using the original format, it is best to define a new (say) altquote
environment for this purpose.
An updated version of quote
allows for an optional argument [name]
which prepends "name said:" before adding the quote. If no name
is supplied, the default quote
(above) is set:

\documentclass{article}
\usepackage{lipsum}% http://ctan.org/pkg/lipsum
\usepackage{ifmtarg}% http://ctan.org/pkg/ifmtarg
\makeatletter
\renewenvironment{quote}[1][]
{\list{}{\rightmargin\leftmargin}%
\item\relax\tiny\@ifmtarg{#1}{\relax}
{\textbf{#1}~said:~}\textquotedblleft\ignorespaces}
{\unskip\unskip\textquotedblright\endlist}
\makeatother
\begin{document}
\lipsum[1]
\begin{quote}
\lipsum*[2]
\end{quote}
\lipsum[3]
\begin{quote}[bkarpuz]
\lipsum*[4]
\end{quote}
\end{document}
Testing for empty arguments is supplied by the ifmtarg
package's conditional
\@ifmtarg{<arg>}
{<Code for arg empty>}
{<Code for arg not empty>}