I understand what you mean about the code, but although it is not OO, everything is well separated, so it's not really "spaghetti".
However, you're right it doesn't follow today's "standards". This is something I'm hoping to change. We already have an autoloader for some classes and I've planned out a new way to do function overrides with classes (e.g. moving util/string.php functions to a StringUtils class). It's hard to do without breaking all existing plugins and making a lot of work for plugin developers!
We're not going to rewrite Q2A on top of Laravel or any other framework. Q2A is a framework in itself so it wouldn't make sense. We can certainly take ideas from those frameworks though.