Would a mini display port to hdmi cable transmit sound to my monitor?

My Lenovo Thinkpad (school laptop) has the worst screen I've ever seen and I need it plugged into an external monitor. It doesn't have an HDMI port and I've been using VGA. It has a mini display port. Would a mini display port to hdmi cable be able to transmit sound to my PC speakers?

Theoretically yes. The mini display port specs do support digital audio, although this may not be fully implemented. Apple in particular do not provide audio via minidisplay port. Check the specs of your Lenovo.

While the DisplayPort specification can support digital audio, the older 2009 line of MacBooks, MacBook Pros, and Mac Minis can't provide an audio signal through the Mini DisplayPort, and only do so over USB, Firewire, or the audio line out port. (The April 2010 line of MacBook Pro, and July 2010 iMac and later do support this[14]). This can be a problem for users who want to connect their computers to HDTVs using a Mini DisplayPort to HDMI adapter. To work around this issue, some third-party manufacturers have created dual or triple-headed adapters that get power for the adapter from a USB port, video from the Mini DisplayPort, and audio from either the USB port or the optical-out port. Either option terminates with a single female HDMI connector, thus allowing both video and audio to be channeled over the single HDMI cable.