No, DVI does not have any means of passing audio through. For that, you will need to get a separate cable and hardware for outputting audio. This can be done using a DAC (like a home theater receiver). Input HDMI from the PS4 into the receiver, output HDMI to DVI to the monitor, and output audio from the receiver to the computer speakers or headphones.
Marantz - 50W 5.2-Ch. 4K Ultra HD and 3D Pass-Through A/V Home Theater Receiver - Black
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