There are many possibilities, see Two figures side by side or LaTeX figures side by side [duplicate].
For example with the floatrow (page 99) package:
\documentclass{article}
\usepackage{caption}
\usepackage[style=plain]{floatrow}
\usepackage{graphicx}
\begin{document}
\begin{figure}[H]
\CommonHeightRow{%
\begin{floatrow}[2]%
\ffigbox[\FBwidth]
{\includegraphics[height=\CommonHeight]{example-image-a}}
{\caption{Energy Vs Photon Count graph for day 44.6}}
\ffigbox[\FBwidth]
{\includegraphics[height=\CommonHeight]{example-image-b}}
{\caption{Energy Vs Photon Count graph for day 45.9}}
\end{floatrow}}%
\end{figure}
\end{document}
Output:

If you want to delete the caption label, you can use the command \caption*{} or put \captionsetup{labelformat=empty} in your preamble:
\documentclass{article}
\usepackage{caption}
\usepackage[style=plain]{floatrow}
\usepackage{graphicx}
%\captionsetup{labelformat=empty}
\begin{document}
\begin{figure}[H]
\CommonHeightRow{%
\begin{floatrow}[2]%
\ffigbox[\FBwidth]
{\includegraphics[height=\CommonHeight]{example-image-a}}
{\caption*{Energy Vs Photon Count graph for day 44.6}}
\ffigbox[\FBwidth]
{\includegraphics[height=\CommonHeight]{example-image-b}}
{\caption*{Energy Vs Photon Count graph for day 45.9}}
\end{floatrow}}%
\end{figure}
\end{document}
Output:
