"Is an HDMI cable all I need to connect my 2nd monitor to the GPU and have it function? "
It depends on what free plugs are still available on your GPU and what kind of connection your new monitor has.
e.g. my KFA2 GTX1080EXOC has 1 HDMI, 3 DPs and 1 DVI-D.
If you used the same GPU and already used 1 HDMI to your older monitor, you could only use either DP or DVI-D to your new monitor.