update(Increment) date in database by 1

That depends on what you're doing and how you're doing it. Show your relevant code where you are doing this "update query" and "after checking the where condition." For instance, are you already retrieving the data from the db and just need to update something?

public function dayorder_shift(Request $request)
{
    $date_shift = $request->input('day_order');
    //dd($date_shift);

    
    // $data = DB::table('table_academic')->where('date_academic','>',$date_shift)->pluck('date_academic');
    // $data->increment('date_academic',1);

   $data= DB::table('table_academic')->where('date_academic', '>',$date_shift)->increment('date_academic',1);
    return redirect('/calendar')->with('success', "Dates succesfully shifted and updated.");
}

increment() only works on numeric field types, not dates.

You'd have to use raw sql for that. Something like

DB::table('table_academic')
    ->where('date_academic', '>', $date_shift)
    ->update([
        'date_academic' => DB::raw('DATE_ADD("date_academic", INTERVAL 1 DAY)')
    ]);

SQLSTATE[22007]: Invalid datetime format: 1292 Incorrect datetime value: 'date_academic'.this error i am getting

what is the column type of date_academic?

date

If it's truly a date field and not a varchar or something, it should work. I do it all the time. I'm using mysql/mariadb though, are you using something different?

https://dev.mysql.com/doc/refman/5.5/en/date-and-time-functions.html#function_date-add