How can I change the indentation in quote and quotation environments and commands?

What about something like that ?

\newenvironment{myquote}%
  {\list{}{\leftmargin=0.3in\rightmargin=0.3in}\item[]}%
  {\endlist}

You can also make the margin size an option:

\newenvironment{myquote}[1]%
  {\list{}{\leftmargin=#1\rightmargin=#1}\item[]}%
  {\endlist}

And use with:

\begin{myquote}{0.3in}
  % ...
\end{myquote}

Use the etoolbox package to patch the \quote command.

\usepackage{etoolbox}
\patchcmd{\quote}{\rightmargin}{\leftmargin 4em \rightmargin}{}{}

(The default definition of \quote is \list{}{\rightmargin\leftmargin}\item\relax.)

I recently released the quoting package which provides a consolidated environment for displayed text as an alternative to quotation and quote. As the main feature, "[f]irst-line indentation is activated by adding a blank line before the quoting environment". With regard to the question at hand: It is also possible to change left-hand and right-hand indentation globally or for single environments using a key--value syntax.

\documentclass{article}

\usepackage[english]{babel}
\usepackage{blindtext}

\usepackage[leftmargin=3em]{quoting}

\begin{document}

\blindtext
% Removing the comment sign in the following line will activate first-line indentation
%
\begin{quoting}
\blindtext
\end{quoting}

\blindtext
%
\begin{quoting}[leftmargin=\parindent]
\blindtext
\end{quoting}

\blindtext

\end{document}