For such jobs, tcolorbox
is more suitable. The advantage here is the box is breakable across pages.
\documentclass{article}
\usepackage{showframe,lipsum} %% just for demo
\usepackage[most]{tcolorbox}
\definecolor{block-gray}{gray}{0.85}
\newtcolorbox{myquote}{colback=block-gray,grow to right by=-10mm,grow to left by=-10mm,
boxrule=0pt,boxsep=0pt,breakable}
\begin{document}
\noindent
\begin{myquote}
\lipsum
\end{myquote}
\end{document}

In your method, you may better use environ
package. However, this is not breakable across pages.
\documentclass{article}
\usepackage{xcolor,showframe}
\definecolor{block-gray}{gray}{0.85}
\usepackage{environ}
\NewEnviron{myblock}
{\colorbox{block-gray}{%
\parbox{\dimexpr\linewidth-2\fboxsep\relax}{%
\small\addtolength{\leftskip}{10mm}
\addtolength{\rightskip}{10mm}
\BODY}}
}
\renewcommand{\quote}{\myblock}
\renewcommand{\endquote}{\endmyblock}
\begin{document}
\noindent
\begin{quote}
The quote come here The quote come here The quote come here The quote come here The quote come hereThe quote come here The quote come here The quote come here The quote come here
\end{quote}
\end{document}
