Okay first, it doesn't matter what 'size' a monitor is (you have a 20inch now wanting to go to 24 inch) only on the display output level it can support and what your video card can output. In this case the video card you have can support more than 1920x1080 (simple google search for that model was the second link here http://www.geforce.com/hardware/desktop-gpus/geforce-9400-gt/specifications). So to answer the basic question, NO you don't need to change video cards.
What everyone is suggesting though, is if your going to go higher level display (1920x1080) especially at 120Hz your probably doing something graphical (gaming, video editing, editing graphics), but many OLDER people want the same thing simply just to see 'things better' and all they use is still Word, Excel, Surfing webpages and the occassional funny Youtube video in full screen mode. If your the latter, then there is still no reason to change your video card.
If your the former, where you want to do graphical, then YES you absolutely need to change the video card, and I would HIGHLY recommend staying with NVIDIA as it supports 120Hz displays (ATI/AMD still has drivers issues it had never really addressed / put resources to resolve the issue). Any level video card your going to get requires HEAVY additional power it can't get from the motherboard slot, so normally you also need to get a Power Supply Unit (PSU) as well, I would recommend AT LEAST 650W up to 800W normally, unless your going real high end on the video card then you may even need a full KiloWatt (yes what your electric bill charges you per minute by) just for the computer (yes your electric bill also goes up significantly). You would need to know how to remove the PSU, install a new one, connect it to all the right connections, and then also swap the video card. Since your not familiar with it, I can recommend some Youtube videos on the subject, but really you may just want to hire the local computer shop to do it for you / local Geek you know to do it.
Doing a replacement Video / PSU INCORRECTLY can FRY your whole computer, cause electrical shocks, and can permanently damage components that would require you to replace before your computer is working again.