\documentclass[twoside,a4paper, 12pt]{article}
\usepackage{fontspec}
\setmainfont[Mapping=tex-text]{Times New Roman}
\newfontfamily\greekfont[Script=Greek]{Times New Roman}
\defaultfontfeatures{Ligatures=TeX}
\usepackage{polyglossia}
\setdefaultlanguage{greek}
\setotherlanguage{english}
\begin{document}
Outside ``english'' environment --

\begin{english}
``Quotes'' -- inside english environment  
\end{english}

\begin{greek}
  ``Ελληνικό'' -- περιβάλλον
\end{greek}

``Ελληνικά''
\end{document}
