Is the quality of sleep dependent on sleep duration?
Sleep duration certainly affects the quality of sleep, but it is nowhere near a simple correlation. It is just one of many factors, and these factors don't work in isolation. Statistically speaking, it's a multivariate model - and one which includes a differential equation, because the way sleep duration affects your sleep today depends on the sleep quality (and also the sleep duration) from yesterday. For example, if you go several day with too little sleep, but keeping firm going-to-bed and waking-up times, the ratio of your sleep phases changes significantly, giving you much more deep sleep. This is what compression therapy is based on. 

And beyond that, the way you feel upon waking is also dependent on sleep quality in very complicated ways, and other factors (for example the synchronisation of the time at which you woke with your circadian cycle) play a much larger role than the sleep quality you got. Also, there are some unexpected effects, where getting too little sleep making you feel good - it's a kind of lightheaded, giddy state, somewhat comparable to having had a small glass of champagne. 

So yes, there is a dependence, but you cannot expect that having had lots of sleep will make you feel good in the morning.