I have a thorough understanding of inner speech, as it was the topic of my Master's thesis. However, I do not have a good recollection of the notion of qualia. Therefore I will focus on the inner speech aspect and hopefully it will indirectly address the connection between the two.
Inner speech definitely has a subjective component and it is also measurable in the brain. As a matter of fact, it lights up nearly precisely the same regions of the brain as actual speech. For instance, the temporal lobe responsible partly for hearing is activated (We imagine that we hear our inner speech, and the brain seems to be fooled into thinking we are really hearing it). This comes as close to being subjective, private conscious experience as one can possibly have without actually measuring consciousness.
In the middle of the 20th century, there were also attempts being made to measure the vocal chords and other speaking apparatuses to see if inner speech and its close cousin "thinking" could be measured by miniscule movements. Some of these studies seemed promising, though fMRI studies decades later far dwarfed these efforts.
Furthermore, the process of voice hearing for schizophrenic patients even lights up similar regions to everyday inner speech of neurotypical people. The only difference is that a person with schizophrenia on some level loses the notion that it is them creating the inner speech, and also loses the ability to predict what the inner speech is going to be. So if a schizophrenic voice could reasonably be defined as a qualia (again, I don't know much about what qualia actually are), the inner speech could also be close to that.
Finally, preliminary studies show that each person has a descriptive for their inner speech. Almost everyone does experience inner speech and some will say it sounds like their own voice, some like a generic nondescript voice, some will say it sounds like different characters such as if they are thinking about Lord of the Rings it may sound British, or if they are saying something deep it may sound like Morgan Freeman.
In my opinion, inner speech does classify as an experience. Some people have particularly rich inner voices, such as writers, academics, and those in contemplative traditions.