How can I make LaTeX number my quotes

One way is to use the resume feature of the enumitem package, which gives you much flexibility in choosing how you want your quotes to be numbered. Below I have defined a list to have a depth of 2 and use numbers for the first level quotes and switch to letters for the second level quotes (if any). The color blue was added just so that the quote environment would stand out:

Notes:

• The showframe package was added just so that you could see where the margins of the text are.

References:

• What are the differences between using paralist vs. enumitem
• Suggestions for transitioning from paralist to enumitem package

Code:

\documentclass{article}
\usepackage{showframe}
\usepackage{lipsum}
\usepackage{xcolor}

\usepackage{enumitem}
\newlist{myQuoteEnumerate}{enumerate}{2}% Set max nesting depth
\setlist[myQuoteEnumerate,1]{label=(\arabic*)}% Use numbers for level 1
\setlist[myQuoteEnumerate,2]{label=(\alph*)}%   Use letters for level 2

\newenvironment{MyQuote}{%
    \begin{myQuoteEnumerate}[resume=*,series=MyQuoteSeries]%
    \color{blue}%
    \item \begin{quote}%
}{%
    \end{quote}%
    \end{myQuoteEnumerate}%
}%

\begin{document}
\noindent
some text some text some text
%
\begin{MyQuote}
    Lorem ipsum dolor sit amet, consectetur adipiscing elit. Mauris et diam viverra mauris commodo eleifend. Donec quis massa nisi.  
    \begin{MyQuote}
        Suspendisse magna nisi, rhoncus sed sagittis sed, auctor id quam. Phasellus quis neque ut tellus tristique accumsan. Curabitur dignissim laoreet pellentesque. Integer placerat consectetur commodo. 
    \end{MyQuote}
\end{MyQuote}
%
more text more text more text
%
\begin{MyQuote}
    Proin elit massa, sollicitudin vitae consectetur sit amet, sollicitudin nec nisl. Nulla facilisi.
\end{MyQuote}
\end{document}

(Edit: please see the additional suggestions made in comments by others.)

Here's a simple approach using a \parbox.

\documentclass[12pt]{article}
\usepackage{lipsum}

\newcounter{quotecount}
\newcommand{\MyQuote}[1]{\vspace{1cm}\addtocounter{quotecount}{1}%
     \parbox{10cm}{\em #1}\hspace*{2cm}(\arabic{quotecount})\\[1cm]}

\begin{document}
\lipsum[1]
\MyQuote{\lipsum[2]}
\lipsum[3]
\MyQuote{\lipsum[4]}
\lipsum[5]

\end{document}