Úplně nerozumím vašemu kódu pouze s těmito informacemi, ale zkuste následující:
Přidejte form
uvnitř smyčky foreach, aby bylo možné odeslat požadavek na jeden objekt Quiz.
Asi takhle:
@foreach($questions as $question)
@if($id == $question->category_id)
<legend> Quiz Of {{ $question->category->title }}</legend>
<div class="jumbotron">
<form method="POST" action="{{ url('/store-quiz') }}" id="quiz_module">
<input type="hidden" name="user_id" value="{{ auth()->user()->id }}">
<input type="hidden" name="category_id" value="{{ $question->category_id }}">
<input type="hidden" name="question_id" value="{{ $question->id }}">
<h4>
Question {{ $question->id }}
</h4>
<h3>
{{ $question->question }}
</h3><br>
<h5>
 <input type="radio" name="user_answer" class="form-check-input" value="{{ $question->option_a }}">{{ $question->option_a }}<br>
 <input type="radio" name="user_answer" class="form-check-input" value="{{ $question->option_b }}">{{ $question->option_b }}<br>
 <input type="radio" name="user_answer" class="form-check-input" value="{{ $question->option_c }}">{{ $question->option_c }}<br>
 <input type="radio" name="user_answer" class="form-check-input" value="{{ $question->option_d }}">{{ $question->option_d }}<br>
</h5>
<button type="submit" class="btn btn-primary">Submit</button>
</form>
</div>
@endif
@endforeach