Is it true that HDMI cables provide the best quality HD picture because the audio and video connections is in the same cable?

There is no truth in the statement.

HDMI is an interface that carries digital audio and video signals. The image quality is dependent entirely on the quality and resolution of the signal that is delivered to the HDMI port by the video source. The HDMI connectors and cable carry that signal to the receiver but the cable does not change it in any way at all.

DVI is compatible with HDMI other than the fact that there is no audio carried with the video signal. The video signal is identical to that carried by an HDMI cable and so this alone proves that statement to be false.