Probably this will cause some other troubles, but here it is.
Elaborating a bit on this answer of Martin Scharrer, this is the result:

Code:
\documentclass{article}
\usepackage[ngerman]{babel}
\let\oldquote'
\newif\ifquoteopen
\catcode`\'=\active
\makeatletter
\DeclareRobustCommand*{'}{%
\@ifnextchar'{%
\ifquoteopen
\global\quoteopenfalse\grqq\expandafter\@gobble
\else
\global\quoteopentrue\glqq\expandafter\@gobble
\fi
}{\oldquote}%
}
\makeatother
\begin{document}
\section{Using ''quotes''}
A ''quote'' and one with a period that follows: ''quote''. And a single 'quote'.
\noindent
And this is the original one: \glqq quote\grqq.
\end{document}
If the meaning of '
in math mode is to be preserved, some other hacks are needed:
\documentclass{article}
\usepackage[ngerman]{babel}
\let\oldquote'
\newif\ifquoteopen
\catcode`\'=\active
\makeatletter
% we have to redefine \pr@m@s to use an active '
\def\pr@m@s{%
\ifx'\@let@token
\expandafter\pr@@@s
\else
\ifx^\@let@token
\expandafter\expandafter\expandafter\pr@@@t
\else
\egroup
\fi
\fi}
\protected\def'{%
\ifmmode
\expandafter\active@math@prime
\else
\expandafter\active@text@prime
\fi}
\def\active@text@prime{%
\@ifnextchar'{%
\ifquoteopen
\global\quoteopenfalse\grqq\expandafter\@gobble
\else
\global\quoteopentrue\glqq\expandafter\@gobble
\fi
}{\oldquote}%
}
\makeatother
\begin{document}
\section{Using ''quotes''}
A ''quote'' and one with a period that follows: ''quote''. And a single 'quote'.
\noindent
And this is the original one: \glqq quote\grqq.
\noindent
Some derivatives $f'(x)+g''(x)$.
\end{document}

This last solution also transforms single quotes into German ones:
\documentclass{article}
\usepackage[ngerman]{babel}
\usepackage{xspace}
\let\oldquote'
\newif\ifquoteopen
\catcode`\'=\active
\makeatletter
% we have to redefine \pr@m@s to use an active '
\def\pr@m@s{%
\ifx'\@let@token
\expandafter\pr@@@s
\else
\ifx^\@let@token
\expandafter\expandafter\expandafter\pr@@@t
\else
\egroup
\fi
\fi}
\protected\def'{%
\ifmmode
\expandafter\active@math@prime
\else
\expandafter\active@text@prime
\fi}
\def\active@text@prime{%
\@ifnextchar'{%
\ifquoteopen
\global\quoteopenfalse\grqq\expandafter\@gobble
\else
\global\quoteopentrue\glqq\expandafter\@gobble
\fi
}{%
\ifquoteopen
\global\quoteopenfalse\grq\xspace
\else
\global\quoteopentrue\glq
\fi
}%
}
\makeatother
\begin{document}
\section{Using ''quotes''}
A ''quote'' and one with a period that follows: ''quote''. A single 'quote' and one with a period that follows: 'quote'.
\noindent
And these are the original ones: \glqq quote\grqq{} and \glq quote\grq.
\noindent
Some derivatives $f'(x)+g''(x)$.
\end{document}

csqoutes
packagecsquotes
.''text''
is not the standard way,``text''
is. But for German I recommend usingbabel
's shorthands"`text"'