\documentclass{article}
\newenvironment{myquote}{\begin{quote} MY\\}{\end{quote}}
\newenvironment{quote1}{\begin{quote} 1\\}{\end{quote}}
\begin{document}
\begin{quote}
A Quote!
\end{quote}

\begin{myquote}
A Quote!
\end{myquote}

\begin{quote1}
A Quote!
\end{quote1}

\renewenvironment{quote1}{\begin{quote} ONE\\}{\end{quote}}

\begin{quote1}
A Quote!
\end{quote1}

\end{document}
