Assuming you want to get all students where any of the qualifications or skills criteria are met:
$skills = $request->skills;
$qualifications = $request->qualifications;
return App\Student::whereHas('qualifications', function ($q) use ($qualifications) {
$q->whereIn('qualification_id', $qualifications);
})->orWhereHas('skills', function ($q) use ($skills) {
$q->whereIn('skill_id', $skills);
})->get();
You will be receiving an array each of qualifications and skills from the form. This assumes you have created the belongsToMany relationships on the App\Student model for both qualifications and skills.
If you wanted to constrain by any selected qualification and any selected skill, change out orWhereHas with whereHas