HiLindaClemons**,**
No, you cannot record a TV program by connecting the TV to the computer using the HDMI cable. You can just connect the TV to the computer for getting the display. For recording the TV program you need to have a TV tuner present on the computer and then you can use Windows media center to record the TV program on the computer itself.
Refer to the below links for further assistance and check -
http://windows.microsoft.com/en-US/windows-vista/Connect-your-computer-to-a-TV
http://windows.microsoft.com/en-US/windows-vista/HDTV-frequently-asked-questions
http://windows.microsoft.com/en-US/windows-vista/Record-TV-in-Windows-Media-Center
http://windows.microsoft.com/en-US/windows-vista/Demo-Watch-pause-and-record-TV-on-your-PC
Hope this helps.