Here is the definition of \alph (and the accompanying \@alph) in the LaTeX kernel:
\def\alph#1{\expandafter\@alph\csname c@#1\endcsname}
\def\@alph#1{%
\ifcase#1\or a\or b\or c\or d\or e\or f\or g\or h\or i\or j\or
k\or l\or m\or n\or o\or p\or q\or r\or s\or t\or u\or v\or w\or x\or
y\or z\else\@ctrerr\fi}
\@alph conditions on the value of the passed counter using an \ifcase...\fi statement. You can create your own counter representation this way. I've done so using \slalph and \@slalph below (with the alphabet taken from Wikipedia's Slovene alphabet):

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[slovene]{babel}
\makeatletter
\def\slalph#1{\expandafter\@slalph\csname c@#1\endcsname}
\def\@slalph#1{%
\ifcase#1\or a\or b\or c\or č\or d\or e\or f\or g\or h\or i\or j\or
k\or l\or m\or n\or o\or p\or r\or s\or š\or t\or u\or v\or
z\or ž\else\@ctrerr\fi}
\makeatother
\renewcommand\thesection{\slalph{section}}
\begin{document}
\section{}
\section{}
\section{}
\section{}
\end{document}
You could do the same with the uppercase alphabet and define (say) \slAlph (and \@slAlph) accordingly.