How can I group all these equations in just one group and write them without the error Missing $ inserted. ...gy}

You are nesting math mode \[ \] in another math environment {equation}. This is a recommended version of your equation.

Some notes:

• The number should go to the last line of the equation.
• Use \text from amsmath to display text.
• Use align from amsmath to align equations on different lines.
• Don't use numbers in labels. Use meaningful labels for \label.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align}
  \text{EQE}&=\frac{\#\,\text{photons}_{\text{out}}}{\#\,\text{electrons}_\text{in}}\nonumber\\
  &=\frac{\text{radiation flux}/\text{photon energy}}{\text{injected current}/\text{elementary charge}}\nonumber\\
  &=\frac{\Phi_e/E_{\text{ph}}}{I_{\text{in}/e}}=\frac{\Phi_e\times e}{I_{\text{in}\times E_{\text{ph}}}}\nonumber\\
  &=\frac{\pi\times A\times L_{\text{v}}\times e\times \lambda_{\text{average}}}{K_{\text{m}}\times V(\lambda_{\text{peak}})\times I_{\text{in}}\times hc}\label{eq:somename}
\end{align}
\end{document}

I would use a single aligned environment inside an equation environment. I would also use \mathrm and \textnormal for words and acronyms.

\documentclass{article}
\usepackage{amsmath} % for 'aligned' environment
\usepackage{newpxtext,newpxmath} % optional
\numberwithin{equation}{section} % optional

\begin{document}
\setcounter{section}{4}  % just for this example
\setcounter{equation}{4}

\begin{equation} \label{eq:EQE}
\begin{aligned}
\mathrm{EQE}
&= \frac{\textnormal{\#photons}_{\textnormal{out}}}{\textnormal{\#electrons}_{\textnormal{in}}}\\
&= \frac{\textnormal{radiant flux}/\textnormal{photon energy}}{\textnormal{injected current}/\textnormal{elementary charge}}
\end{aligned}
\end{equation}
\end{document}