The way that organic search engine results and paid placements are displayed works separately on Google - the two systems do not interact!
How does AdWords Quality Score effect position and how much you pay? A beginning advertiser might think that the bid they set in AdWords is the price they'll pay each time someone clicks their ad. They'd be dead wrong. Your quality score determines where your ad will rank on the page, how much it will cost.
First AdWords determines where your ad will rank of the page by multiplying your Maximum CPC by the keyword QS.
Second, AdWords determines your actual CPC by dividing the ad rank of the ad beneath yours by your keyword QS and rounding up to the nearest cent.
How does your landing page effect keyword Quality Score?
Landing page relevance has been a factor in QS since 2005. However, increasing the organic ranking of a page will have no effect on your quality score. You could have 10,000 high quality backlinks to your paid search landing page and it wouldn't increase your quality score. Anyone who tells you otherwise doesn't know any better, or is likely trying to upsell you on SEO (Let me preface I'm not against SEO, it's definitely worthwhile just not relevant to determining quality score).
The landing page portion of QS is determined by page relevance. Google checks to make sure your landing page fulfills the promise of the ad. If you promise basket balls in the ad and the page talks about dresses you have a relevance issue. Intuitively this makes sense, if people start to feel that paid search ads aren't relevant they'll be less likely to click them, causing Google to lose money.
AdWords QS is also determined by originality of on page content, which is probably where people get hung up thinking that quality score is linked to SEO (unique content is important for ranking well in organic results). When in AdWords land this just means you shouldn't be using mirror or bridge pages that don't provide value to the visitor.
Another piece that factors into the landing page portion of QS is page time load speeds. A page that doesn't load quickly is a poor experience for the visitor.
For more about how landing pages effect quality score check out Google's official stance on the topic.
While landing pages are important, they only effect about 20% of your quality score. To improve quality score you're much better off focusing on creating a proper account structure, correct match types for keywords and focusing on Click Through Rates. If anything, it should be simple to make your landing page relevant to quality score. It's much more important to focus on making a landing page that gets those hard earned clicks to take an action, but that is better left for a future discussion.