How to color just the `\"`?

Perhaps just wrap it in a macro, like so:

\documentclass{article}
\usepackage{mathtools,xcolor}

\newcommand{\oo}{\makebox[0pt][l]{\color{red}\"{}}o}
\newcommand{\cc}[1]{\makebox[0pt][l]{\color{red}\c{#1}}#1}

\begin{document}
    L\oo\oo{k} at this: \cc{c} and \cc{s}
\end{document}

You could use stackengine:

\documentclass{article}
\usepackage{mathtools,xcolor}
\usepackage{stackengine}

\newcommand{\my}[2]{\stackinset{c}{}{c}{1.1pt}{\color{red}#2}{\color{black}#1}}
\newcommand{\myc}[1]{\stackinset{c}{}{c}{.8pt}{#1}{\color{red}\c{#1}}}

\begin{document}
    S\my{o}{\"}\"{o}mething \c{c}\myc{c}o\c{s}\myc{s}a
\end{document}

Accents over nothing are raised by 1ex:

\documentclass{article}
\usepackage{xcolor}

\newcommand{\cu}[2][red]{% colored umlaut
  \leavevmode
  \vbox{
    \halign{%
      \hfil##\hfil\cr
      \textcolor{#1}{\"{}}\cr
      \noalign{\nointerlineskip\kern-1ex}
      #2\cr
    }
  }%
}
\newcommand{\cc}[2][red]{% colored cedilla
 \leavevmode
 \begingroup
 \ooalign{\hfil#2\hfil\cr\hfil\textcolor{#1}{\c{}}\hfil\cr}%
 \endgroup
}

\begin{document}

\cu{O}zg\cu{u}r \"Ozg\"ur \cc[blue]{C}\cc{c}\cc{S}\cc{s} \c{C}\c{c}\c{S}\c{s}

\bigskip

\LARGE \cu{O}zg\cu{u}r \"Ozg\"ur

\end{document}