Typically no. It is a video display interface only. However, there are some computer graphics cards and sources that do support output of audio using non-standard extensions to the spec via the DVI-D port and a DVI-to-HDMI adapter or cable. These are readily available.
There are also DVI + R/L stereo audio-to-HDMI converters that take in the DVI video plus right and left stereo output from the sound card to combine them into an HDMI signal containing both the video and audio. Usually used to connect an older computer with DVI output video card and sound card audio output to an HDTV with HDMI input. These are powered converters, but relatively inexpensive and readily available.