\documentclass{standalone}
\usepackage{pgfplots}
\usetikzlibrary{intersections}
\begin{document}

\newcommand*{\ShowIntersection}[2]{
\fill 
    [name intersections={of=#1 and #2, name=i, total=\t}] 
    [red, opacity=1, every node/.style={black, opacity=1}] 
    \foreach \s in {1,...,\t}{(i-\s) circle (2pt)
        node [above right] {\s}};
}

\begin{tikzpicture}
\begin{axis}[xmin=-1.5, xmax=1.5,
    ymin=1, ymax=3,
    x axis line style={name path=AxisLine}]

\addplot[name path global=a, mark=none, domain=-2.5:2.5, thick]%
    ({x},{x*x});%
\ShowIntersection{a}{AxisLine}

\end{axis} 
\end{tikzpicture}
\end{document}
