I'll follow up with some references below, but first consider these ideas that I, arguably a philosopher, would use to try to convince someone that qualia exist:
- at the day to day level we seem to experience and talk about qualia,
- the mental phenomena referred to are sufficiently distinct from the raw sensory experience that they are conceptually different things.
therefore, for some levels of discussion, qualia exist.
There is empirical support for the existence of qualia: my own mental states usually include features like "perceiving red" etc. I suspect that you find the same to hold for your mental states. In addition people in general report about their mental states in a way that is consistent with all of them having qualia. Thus at the common-sense day to day (i.e. pre-theoretic) level, qualia exist. The main question is whether these apparent qualia are in some way fundamental, or otherwise irreducable to other mental or physical characteristics. (The idea that objects have an intrinsic tendency to slow down in Aristotles' physics seems like a rough analog. everybody sees that this is the case in their day to day lives, but it is not an irreducable feature of a more accurate theory)
Another empirical fact is that our qualia are influenced by things other than what would seem to be the most obvious physical determinant: our perception of color depends on what is around a given object (numerous optical illusions depend on this), our perception of volume depends on pitch, memory can play a role (the "color-reversals after staring" type of optical illusions), people literally "seeing red" when angry etc. These features indicate that, at the very least, the connection between physical stimuli and the resulting qualia is very complex; perhaps so complex that it is worth identifying qualia as being conceptually distinct entities from the stimuli.
This level of discussion does not pin down the features of qualia in sufficient detail to be considered a theory of the phenomenon, however, it indicates that qualia need to be addressed, in one way or another, in any complete theory of mind.
Edit: Additional Reference
Although he does not explicitly reference qualia, Thomas Nagel follows essentially this line of thought in What Is It Like to Be a Bat?: humans, and bats, and other organisms have subjective experiences, it's unclear how these relate to physical phenomena, thus there is a problem in reductionistic approaches to mind. Namely, they do not address this issue of qualia -- how qualia arise, what their connection to the physical is etc. His own proposal for addressing this is to more completely, and objectively, pin down the features/characteritics of qualia (my interpretation), in his words "we can pursue a more objective understanding of the mental in its own right", and only by doing so can we resolve questions about physicalism.
As far as I can tell, David Chalmers takes it as self evident: e.g. "It is undeniable that some organisms are subjects of experience." (That is, I'm unaware of any explicit argument aimed at convincing someone who is skeptical of the idea of qualia).