Quality score's BIGGEST factor is their CTR which usually is a good indicator of relevancy (as the user would be more compelled to click if the ad matches their query). Depending on what you're current CTR is relative to what Google thinks your CTR should be, I bet raising just that alone would boost your quality score for that keyword.
I do PPC for a living working in education industry so it's pretty damn competitive and we noticed just working on increasing the CTR is enough to increase quality score.
To be honest, Google only looks at landing pages to make sure it's relevent enough, as long as the bubble next to your status says "Landing page quality: No Problem", you're golden (this is from Google Adwords Evangelist, Frederick Vallaeys back in 2010).
Hope this helps, if you want to tell me your keyword, PM me and I might be able to give some insight.