To achieve vertical centering of the material in the c-type columns, I suggest you execute
\renewcommand\tabularxcolumn[1]{m{#1}}
This is explained in more detail in the user guide of the tabularx package.
I'd also suggest you complete the \multirow commands and -- in the header row -- change \textbf{$q_i$} to \boldmath$q_i$ and change \textbf{Definition of $z_i$} to \boldmath\bfseries Definition of $z_i$, respectively. And, please change $q_i$ = 1 to $q_i = 1$, etc in column 2. I'd further suggest to get rid of the specious \centering instruction and change \captionof{table}{...} to just \caption{...}. (There's no need for \captionof since you're using a table environment.) And, last but not least, consider adding the instruction \setlength\extrarowheight{2pt} to that the table gets a bit less of a compressed "look".

\documentclass[journal]{IEEEtran}
\usepackage{caption}
\usepackage{tabularx}
\renewcommand\tabularxcolumn[1]{m{#1}} % <-- new
\usepackage{multirow}
\begin{document}
\begin{table}[h]
\setlength\extrarowheight{2pt}
\caption{Table of $q$ and $z$}
\label{tab:table_q_and_z}
\noindent
\begin{tabularx}{\columnwidth} { | c | c | >{\raggedright\arraybackslash}X |}
\hline
\textbf{Steps} & \boldmath$q_i$ & \boldmath\bfseries Definition of $z_i$ \
\hline
\multirow{3}{}{$i \leq (n-1)$}
& $q_i=0$
& $z_i$ = number of \emph{emptied} chambers at the \emph{cleaning} place $i \leq (n-1)$ \
\cline{2-3}
& $q_i=1$
& $z_i$ = number of \emph{loaded} chambers at the \emph{processing} place $i \leq (n-1)$ \
\hline
\multirow{3}{}{$i=n$}
& $q_n=0$
& $z_n$ = number of \emph{loaded} chambers at the \emph{processing} place $n$ \
\cline{2-3}
& $q_n=1$
& $z_{n}$ = number of \emph{emptied} chambers at the \emph{cleaning} place $n$ \
\hline
\end{tabularx}
\end{table}
\end{document}
Addendum to address some additional issues with the table:
The IEEEtran document class is only barely compatible with the caption package. Don't use this package unless the journal you are planning to submit a paper to expressly allows you to use this package.
The IEEEtran class uses Times Roman as the default text font, but Computer Modern as the default math font. These two fonts are not all that well matched, visually/aesthetically speaking. I'd recommend executing \usepackage{newtxtext,newtxmath} in the preamble to use Times Roman for both text and math.
Consider getting rid of the all verical lines and most horizontal lines in the table, in order to give the table a much more open and inviting "look". I suggest you use the line-drawing macros of the booktabs package instead of \hline and \cline.
Using bold-facing in the header row doesn't achieve much, except run the risk of giving the entire table a heavy-handed and potentially vulgar look. Don't over-use bold-facing; for the table at hand, I don't think any bold-facing is needed.
Implementing these ideas, dropping vertical centering in columns 1 and 2, and left-aligning the material in columns 1 and 2, would create the following look. I hope you will agree with me that the table looks a whole lot more inviting and appealing than the one posted above.

\documentclass[journal]{IEEEtran}
\usepackage{tabularx}
\usepackage{newtxtext,newtxmath}
\usepackage{booktabs} % for well-spaced horizontal rules
\usepackage{lipsum} % filler text
\begin{document}
\begin{table}[h]
\caption{Table of $q$ and $z$}
\label{tab:table_q_and_z}
\begin{tabularx}{\columnwidth} { @{}
lll
@{\hspace{\tabcolsep}}
>{\raggedright\arraybackslash}X
@{}}
\toprule
Steps & $q_i$
& \multicolumn{2}{l}{Definitions of $z_i$ and $z_n$}\
\midrule
$i\leq n-1$ & $q_i=0$ & $z_i$
& number of \emph{emptied} chambers at the \emph{cleaning} place $i \leq (n-1)$ \
& $q_i=1$ & $z_i$
& number of \emph{loaded} chambers at the \emph{processing} place $i \leq (n-1)$ \
\addlinespace
$i=n$ & $q_n=0$ & $z_n$
& number of \emph{loaded} chambers at the \emph{processing} place $n$ \
& $q_n=1$ & $z_n$
& number of \emph{emptied} chambers at the \emph{cleaning} place $n$ \
\bottomrule
\end{tabularx}
\end{table}
\lipsum
\end{document}
IEEEtranclass uses Times Roman as the default text font, you may want to consider executing\usepackage{newtxtext,newtxmath}in order to get a compatible Times Roman math font. – Mico Apr 02 '23 at 11:15