If you don't have to use minted, you could try my pythontex package, which uses the same Pygments highlighting library. It provides a \pygment command for inline use, and this seems to work fine in footnotes, except that # and % characters don't work (the # and % work fine outside footnotes in normal text, and it wouldn't be hard to get around the footnote issue if these characters are needed). The package also provides a pygments environment and an \inputpygments command.
\documentclass[a4paper,10pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[italian]{babel}
\usepackage{pythontex}
\begin{document}
\footnote{Quotes: \pygment{bash}/""/. More: \pygment{bash}/echo ${a[$i,$j]}/.}
\end{document}

More explanation and patch for minted
The ultimate issue is that minted uses verbatim content (ShortVerb from fancyvrb), and that's problematic in footnotes. My package uses a different approach, and also caches all highlighted content to increase speed.
Here's a rough patch for minted that should give you what you want. It makes \mint into an inline command. It's based on some things I did in my package, so it should work in footnotes as well as my package does. I had to make new versions of \minted@savecode and \minted@pygmentize to get rid of excess spaces (added lots of %), and to temporarily swap Verbatim for BVerbatim. It's very important to avoid any extra spaces hanging around for inline use; the examples show that regular text can be right up against highlighted content.
Comment on performance: If you are going to redefine \mint for inline use (or, perhaps better, define a new command for that purpose), you will presumably use inline code fairly frequently. On my system, compiling takes about 12 s when \mint{bash}/""/ is used 40 times; it's about 23 s for 80 times. pythontex takes about 3.5 s for 40 uses of \pygment{bash}/""/ (this involves running pdflatex, then the pythontex script, and then pdflatex again, to bring back highlighted results). It's about the same for 80 uses, about 5.5 s for 800 uses, and about 23 s for 8000. And all of those numbers are only when pythontex actually has to highlight everything; since results are saved, the pythontex script and second pdflatex run are only needed when code is modified. If you are recompiling after editing non-code text, you only need to run pdflatex, and should get more or less normal pdflatex speeds.

\documentclass[a4paper,10pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[italian]{babel}
\usepackage{minted}
\makeatletter
\renewcommand\mint{%
\begingroup
\let\do\@makeother\dospecials
\catcode`\{=1
\catcode`\}=2
\@ifnextchar[{\endgroup\mint@i}{\endgroup\mint@i[]}}
\def\mint@i[#1]#2{%
\minted@resetoptions
\setkeys{minted@opt}{#1}%
\gdef\mint@lang{#2}%
\begingroup
\let\do\@makeother\dospecials
\mint@ii}
\def\mint@ii#1{%
\endgroup
\def\mint@iii##1#1{%
\endgroup
\expandafter\edef\csname FV@SV@minted@verb\endcsname{\detokenize{##1}}%
\minted@savecode@inline{\FV@SV@minted@verb}%
\minted@pygmentize@inline{\mint@lang}%
\DeleteFile{\jobname.pyg}}%
\begingroup
\let\do\@makeother\dospecials
\mint@iii}
\newcommand\minted@savecode@inline[1]{%
\immediate\openout\minted@code\jobname.pyg
\immediate\write\minted@code{#1}%
\immediate\closeout\minted@code}
\newcommand\minted@pygmentize@inline[2][\jobname.pyg]{%
\RecustomVerbatimEnvironment{Verbatim}{BVerbatim}{}%
\def\minted@cmd{pygmentize -l #2 -f latex -F tokenmerge
\minted@opt{gobble} \minted@opt{texcl} \minted@opt{mathescape}
\minted@opt{startinline} \minted@opt{funcnamehighlighting}
\minted@opt{linenos} -P "verboptions=\minted@opt{extra}"
-o \jobname.out.pyg #1}%
\immediate\write18{\minted@cmd}%
% For debugging, uncomment:
%\immediate\typeout{\minted@cmd}
\ifthenelse{\equal{\minted@opt@bgcolor}{}}%
{}%
{\begin{minted@colorbg}{\minted@opt@bgcolor}}%
\input{\jobname.out.pyg}%
\ifthenelse{\equal{\minted@opt@bgcolor}{}}%
{}%
{\end{minted@colorbg}}%
\DeleteFile{\jobname.out.pyg}}
\makeatother
\begin{document}
Before environment
\begin{minted}{bash}
""
\end{minted}
After environment
Before inline\mint[]{bash}/""/After inline
Before note\footnote{Before\mint{bash}/""/After}
\end{document}
Edit 2013/07/30
I am now maintaining minted, and have incorporated code similar to the patch here to create a new inline command, \mintinline, in the development version on GitHub.
\shorthandoff{"}after\begin{document}, if you don't need the shorthands. However, expect problems with\mintin footnotes. – egreg Nov 20 '12 at 18:44mintin footnotes could create problems? And, since I need to have some syntax highlighting of keywords in the footnotes how should I do it without usingmintand being consistent with the rest of the document? (Also note that I cannot use custom commands otherwise Kile would screw up the highlightining due to a lot of $ inside code, so saying create a newcommand with an optional arg is not a solution). – Bakuriu Nov 20 '12 at 19:44