Equation number with an apostrophe

a method that will allow both equation numbers to be referenced outside the align uses labels and \ref:

\begin{align}
 x^2+y^2=z^2, \label{firsteq}\\
 x^2=7, \tag{\ref*{firsteq}'} \label{primeq}
\end{align}

the job needs to be run three times to resolve the numbers.

the * on \ref* in the \tag{...} will prevent the equation number in the display from being treated as a link. however, the label for the number with the prime can still be referenced in text by \eqref, and that will be linked.

(not part of the answer, but a suggestion: prime is more usual than an apostrophe; the apostrophe in this context will automatically be set as a prime.)

I would recommend using the \tag macro to directly insert the equation number.

\documentclass{article}
\usepackage{amsmath,amssymb}
\numberwithin{equation}{section}
\begin{document}

\section{Section A}

\begin{align}
 z^2&= x^2 + y^2     \\
 z^2 &=  x^2 + y^2  \tag{\theequation'}
\end{align}

\end{document}

As per @barbarabeeton 's answer, the ' should be \prime as:

\documentclass{article}
\usepackage{amsmath,amssymb}
\numberwithin{equation}{section}
\begin{document}

\section{Section A}

\begin{align}
 z^2&= x^2 + y^2     \\
 z^2 &=  x^2 + y^2  \tag{\theequation${}^\prime$}
\end{align}

\begin{equation}
 z^2= x^2 + y^2     
\end{equation}
\end{document}

Incidentally, the \theequation is the way to access directly the formatted equation counter. This works with any counter you're using, \the<counter_name> will call the formatted counter. This is the reason \theequation resulted in 1.1 and not just 1, which is the value of the counter equation.