The "trick" here is that, for a video card to provide both video and audio on one HDMI cable, the video card must have its own sound chip on board. Most modern cards do, but check the specs of what you plan to buy.
THEN there's another step you must do, assuming your new card does have audio output on its HDMI connector. Once it is installed and you have installed the card's driver software, you must go into Control Panel ... Sounds / Audio Devices and click on the Audio tab. In the Sound Playback section click on the down-arrow to show all the audio playback devices in your machine, and you'll find a new one - the audio system in your new video card. You must set it as your normal audio output device. You see, Windows can only output audio from ONE device, so you need to be using the one in your new video card so that it can put the audio on your HDMI cable.
The video card has no way of getting audio signals from the mobo's chip to relay out the HDMI port - it MUST use its own on-board chip, so you must tell Windows to use that as the audio output device.