Why can't sectsty change the chapter font when it's used with babel package?

The magyar option seizes the initiative and changes the code for \@chapter to reflect the original code in the class and ignores the settings by sectsty. It does nothing to \@schapter, which is the reason you see sans serif in unnumbered chapters.

Use titlesec, instead:

\documentclass[a4paper]{report} 

\usepackage[sf,bf]{titlesec}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}

\newcommand\magyarOptions{chapterhead=unchanged}
\usepackage[magyar]{babel}

\usepackage{blindtext}

\renewcommand{\sfdefault}{qhv}

\titleformat{\chapter}[display]
  {\sffamily\bfseries\huge}
  {\thechapter. \chaptername}
  {20pt}
  {}

\begin{document}

\chapter{Example chapter} %% this one is not working correctly

\blindtext

\section{Example section}

\blindtext

\subsection{Example subsection}

\blindtext

\chapter*{Example unnumbered chapter}

\blindtext

\end{document}

Use \def\magyarOptions{chapterhead=unchanged} before \usepackage[magyar]{babel}

Code:

\documentclass[a4paper]{report}
\def\magyarOptions{chapterhead=unchanged}
\usepackage[magyar]{babel}
\usepackage[utf8]{inputenc}
\usepackage{t1enc}

\usepackage{blindtext}

\usepackage{sectsty}
\chapterfont{\usefont{T1}{qhv}{b}{n}\selectfont\huge} 
\sectionfont{\usefont{T1}{qhv}{b}{n}\selectfont\LARGE}
\subsectionfont{\usefont{T1}{qhv}{b}{n}\selectfont\Large}

\begin{document}
    \chapter{Example chapter} %% this one is not working correctly
    \blindtext
    \section{Example section}
    \blindtext
    \subsection{Example subsection}
    \blindtext
    \chapter*{Example unnumbered chapter}
    \blindtext
\end{document}