Intertwining two letters in TikZ

Not a general solution, but you could just draw the black letter twice, but clip it the second time so that only the part that you want is drawn:

\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
  \fill[gray!60] (0,0) circle (2);
  \node[color=black!80, font=\fontsize{60}{20}\fontfamily{ppl}\fontseries{b}\selectfont]
    at (0.25,-0.25) {C};
  \node[color=yellow!50, font=\fontsize{60}{20}\fontfamily{ppl}\fontseries{b}\selectfont] 
    at (-0.25,0.25) {Q};
  \clip (0,0) -- (0:2) arc (0:90:2) -- cycle;
  \node[color=black!80, font=\fontsize{60}{20}\fontfamily{ppl}\fontseries{b}\selectfont]
    at (0.25,-0.25) {C};
\end{tikzpicture}
\end{document}

This is just a minor addendum to David Purton's nice answer. Some may feel more comfortable just specifying how much of the character should be cut. As long as this is some distance from either of its boundaries, trimclip can do the job (and in principle you would not need tikz then). And it is usually a good idea to use the font or node font key to set the font, and circle[radius=2] instead of circle(2).

\documentclass[tikz,border=3mm]{standalone}
\usepackage{trimclip}
\begin{document}
\begin{tikzpicture}[node font=\fontsize{60}{20}\fontfamily{ppl}\fontseries{b}\selectfont]
 \fill[gray!60] (0,0) circle[radius=2];
 \node[color=black!80,below] at (0.25,1.25) {C};
 \node[color=yellow!50] at (-0.25,0.25) {Q};
 \node[color=black!80,below] at (0.25,1.25) {\clipbox{0em 0.5em 0em 0em}{C}};
\end{tikzpicture}
\end{document}