While the value of the questions-counter is 0 the sequence \setq{\arabic{questions}}{Number}{\arabic{questions}} yields defining the control-word-token \qa:0.Number to deliver the token-sequence \arabic{questions}.
That token-sequence in turn will be carried out/expanded not at the time of defining but at the time right after using/expanding the control-word \qa:0.Number, yielding
the value of the question-counter current not at the time of defining but at the time of using/expanding the control-word \qa:0.Number.
You probably wish to have the sequence \arabic{questions} expanded at the time of defining \qa:0.Number.
Instead of \arabic{questions} I use \number\value{questions} because with the latter you need only one \expandafter-chain for triggering expansion until obtaining the value while with the further you need four \expandafter-chains:
\arabic=macro:
#1->\expandafter \@arabic \csname c@#1\endcsname .
l.16 \show\arabic
? i
insert> \show@arabic
> @arabic=macro:
#1->\number #1.
<insert> \makeatletter\show@arabic
l.16 \show\arabic
The first \expandafter-chain to obtain \expandafter\@arabic\csname c@questions\endcsname from \arabic{questions}.
The second \expandafter-chain to obtain \@arabic\c@questions from \expandafter\@arabic\csname c@questions\endcsname.
The third \expandafter-chain to obtain \number\c@questions from \@arabic\c@questions.
The fourth \expandafter-chain to obtain the sequence of decimal digits denoting the value of the \count-register \c@questions from \number\c@questions.
On my system the following MWE compiles without errors:
\documentclass{article}
% \usepackage[utf8]{inputenc}
\usepackage{totcount}
% \usepackage{pgffor}
\newcommand\PassFirstToSecond[2]{#2{#1}}
\newcommand\setq[3]{%
\global\csname @namedef\endcsname{qa:#1.#2}{#3}%
}
\newtotcounter{questions}
\newcommand\newq[4]{%
\expandafter\PassFirstToSecond\expandafter{\number\value{questions}}{\setq{\arabic{questions}}{Number}}%
\setq{\arabic{questions}}{Date}{#1}%
\setq{\arabic{questions}}{Time}{#2}%
\setq{\arabic{questions}}{Question}{#3}%
\setq{\arabic{questions}}{Answer}{#4}%
\stepcounter{questions}%
}
\newq{268}%
{10:25}%
{Can you create a latex project with multiple documents?}%
{Yes, at least you can in Overleaf. You click on the menu in the top-left, and change main document to the tex file you want to compile.%
}
\begin{document}
\par\noindent
\csname qa:0.Number\endcsname\
\csname qa:0.Date\endcsname\
\csname qa:0.Time\endcsname\
\csname qa:0.Question\endcsname\
\csname qa:0.Answer\endcsname
\end{document}

You might be interested in the \name-macro. I elaborated on the \name-macro in the thread "Define a control sequence after that a space matters" which was started at TeX - LaTeX StackExchange in November 10, 2016 :
\documentclass{article}
% \usepackage[utf8]{inputenc}
\usepackage{totcount}
% \usepackage{pgffor}
\makeatletter
\newcommand\MyMacroNamePrefix@Exchange[2]{#2#1}%
@ifdefinable\name{%
\long\def\name#1#{\romannumeral0\MyMacroNamePrefix@innername{#1}}%
}%
\newcommand\MyMacroNamePrefix@innername[2]{%
\expandafter\MyMacroNamePrefix@Exchange\expandafter{\csname#2\endcsname}{ #1}%
}%
\makeatother
\newtotcounter{questions}
\newcommand\newq[4]{%
\name\name{@ifdefinable}{qa:\arabic{questions}.Number}{%
\name\edef{qa:\arabic{questions}.Number}{\arabic{questions}}%
}%
\name\newcommand{qa:\arabic{questions}.Date}{#1}%
\name\newcommand{qa:\arabic{questions}.Time}{#2}%
\name\newcommand{qa:\arabic{questions}.Question}{#3}%
\name\newcommand{qa:\arabic{questions}.Answer}{#4}%
\stepcounter{questions}%
}%
\newq{268}%
{10:25}%
{Can you create a latex project with multiple documents?}%
{Yes, at least you can in Overleaf. You click on the menu in the top-left, and change main document to the tex file you want to compile.%
}
\begin{document}
\par\noindent
\name{qa:0.Number}\
\name{qa:0.Date}\
\name{qa:0.Time}\
\name{qa:0.Question}\
\name{qa:0.Answer}
\end{document}
