The package xgreek
explicitly redefines \@alph
and \@Alph
to use greek numbers.
If those specific greek numbering style is not needed in the rest of the document, it's possible to switch back to the Latin style by catching the original definition and return to them after xgreek
has been loaded.
\documentclass[a4paper,12pt]{article}
\makeatletter
\let\latex@alph\@alph
\let\latex@Alph\@Alph
\makeatother
\usepackage{unicode-math,xgreek}
\defaultfontfeatures{Mapping=tex-text}
\usepackage{alphalph}
\renewcommand*{\thefootnote}{\alph{footnote}}
\setromanfont[Mapping=tex-text]{Arial Black}%
%GFS Didot}
\usepackage{polyglossia}
\setdefaultlanguage{english}
\setotherlanguage[variant=modern]{greek}
%\alphalph{\value{footnote}}
\makeatletter
\let\@alph\latex@alph
\let\@Alph\latex@Alph
\makeatother
\begin{document}
\section{Foo}
\alph{section}
some text\footnote{test}
\end{document}
Improved version
\documentclass[a4paper,12pt]{article}
\usepackage{unicode-math,xgreek}
\defaultfontfeatures{Mapping=tex-text}
\usepackage{alphalph}
\setromanfont[Mapping=tex-text]{Arial Black}%
%GFS Didot}
\usepackage{polyglossia}
\setdefaultlanguage{english}
\setotherlanguage[variant=modern]{greek}
\makeatletter
\renewcommand*{\thefootnote}{%
\latin@alph{\c@footnote}%
}
\makeatother
\begin{document}
\section{Foo}
\alph{section}
some text\footnote{test}
\end{document}