I think it might be easier to instead redefine the quote environment to take an optional argument, so that it could be used as
\begin{quote}
...
\end{quote}
or
\begin{quote}[Title goes here]
...
\end{quote}
So, in the code below, that's exactly what I've done, using
\let\oldquote\quote
\let\oldendquote\endquote
and then, borrowing the 'empty' test from What is the difference between \empty and \@empty? to redefine the quote environment as
\renewenvironment{quote}[1][]{\oldquote%
\if\relax\detokenize{#1}\relax
``
\else
\begin{center}%
``{\bfseries\itshape #1}%
\end{center}\nobreak
\fi}{''\oldendquote}

Here's the complete MWE
\documentclass{article}
\let\oldquote\quote
\let\oldendquote\endquote
\renewenvironment{quote}[1][]{\oldquote%
\if\relax\detokenize{#1}\relax
``
\else
\begin{center}%
``{\bfseries\itshape #1}%
\end{center}\nobreak
\fi}{''\oldendquote}
\begin{document}
\begin{quote}
Qvidam posuit unum par cuniculorum in quodam loco, qui erat undique
pariete circundatus, ut sciret, quot ex eo paria germinarentur in un
Qvidam posuit unum par cuniculorum in quodam loco, qui erat undique
pariete circundatus, ut sciret, quot ex eo paria germinarentur in un
\end{quote}
\begin{quote}[Here's one with a title]
Qvidam posuit unum par cuniculorum in quodam loco, qui erat undique
pariete circundatus, ut sciret, quot ex eo paria germinarentur in un
Qvidam posuit unum par cuniculorum in quodam loco, qui erat undique
pariete circundatus, ut sciret, quot ex eo paria germinarentur in un
\end{quote}
\end{document}
\begin{center}``\textit{\textbf{#1}}\end{center}? – Marc van Dongen Mar 10 at 4:58\centerlineis a TeX command and it does what it states. It centers the text on one line. Go with Marc's suggestion. – Yiannis Lazarides Mar 10 at 5:52