I’m assuming you have selected the right input on the TV.
Few things, are you sure the graphics card is working? and for that matter, are you sure the G5 is working as well (does it POST?)
Let’s go back to basics:
Once connected, and powered on, do you hear the fans?
There are two versions of DVI. DVI-A and DVI-D (A for Analogue and D for Digital). There is a third version called DVI-I (which is both digital and analogue) but you need the Digital version to be able to connect to a HDMI device (HDMI is pure digital). It’s possible the video cards are in analogue mode as they were previously set but the adapter should give the video card the info as well. As the pin outs on the DVI-A and DVI-D cables are slightly different.
More info is needed.