Using \restoregeometry in environment, next page runs off the page bottom

You're using a group too many and don't restore properly the geometry, as the \restoregeometry command is issued when LaTeX is still processing the environment, which forms a group:

\newenvironment{quotepage}[1]
  {\newpage
   \newgeometry{margin=0pt}
   \thispagestyle{empty}
   \centering
   \vspace*{\fill}\vspace{-\baselineskip}
   \begin{minipage}{#1}
   \setlength{\parskip}{0.6\baselineskip}
   \setlength{\parindent}{0pt}
  }
  {\end{minipage}
   \vfill
   \clearpage
   \aftergroup\restoregeometry
  }

You may want to look at other solutions, for example TikZ provides the ability to put nodes at the physical page center.

\usepackage{tikz}
\newsavebox{\quotepagebox}
\newenvironment{quotepage}[1]
  {\begin{lrbox}{\quotepagebox}\begin{minipage}{#1}
   \setlength{\parskip}{0.6\baselineskip}
   \setlength{\parindent}{0pt}}
  {\end{minipage}\end{lrbox}%
   \clearpage\thispagestyle{empty}
   \begin{tikzpicture}[remember picture,overlay]  
   \node at (current page.center) {\usebox{\quotepagebox}};
   \end{tikzpicture}
   \clearpage}

As egreg said, \restoregeometry must be executed outside the environment.

You can use the \AfterEndEnvironment-hook from the etoolbox package for that.

From the etoolbox documentation:

\AfterEndEnvironment{environment}{code}

Appends arbitrary code to a hook executed at a very late point by the \end command, after the group holding the environment has been closed.

MWE:

\documentclass[a6paper,landscape]{article}
\usepackage{etoolbox}
\usepackage[showframe]{geometry}
\usepackage{lipsum}

\newenvironment{myenv}{Hello!}{Bye!}
\BeforeBeginEnvironment{myenv}{\newgeometry{left=5mm,right=5mm,top=5mm,bottom=5mm}}
\AfterEndEnvironment{myenv}{\restoregeometry}

\begin{document}
\lipsum[1]
\begin{myenv}
Test
\end{myenv}
\lipsum[2-3]
\end{document}

Output: