In my experience, it's not the monitor so much as the VGA signal path is inherently less precise than the digital signal path. You really want to go digital end-to-end.
Failing that, I suggest that you remove the VGA to HDMI converter from the signal path entirely. Every signal conversion causes a loss of precision. If that particular one is worse than usual, try letting the monitor do it.
Specifically, if that card has a third digital output use it (see how lazy I am, I didn't look it up). If not, attach the VGA output directly to that monitor's VGA input (has it got one?). See if that looks any better. Try the VGA input on the other three monitors. If any direct VGA setup gives you a good enough picture, keep that monitor on the VGA port. If not, time for a new card.
@Phillip Corcoran
Yes, age can degrade one's sensitivity to the difference. But many people simply don't see the difference at all; those people are happier because it doesn't bother them. The OP is describing a case that I suspect is fairly unsubtle.
But I'm going to re-tell a story to illustrate just how peoples' eyes differ in sensitivity to this. I once had an AB switchbox at work. Gave me a headache, with a one-pixel offset ghost image. So someone who couldn't see the difference traded me for his switchbox. Same issue. I called the manufacturer of the box; they had never heard of this.
They called me back the next day. EVERY box in their lab did the same thing. No-one had noticed before. They used their own switchboxes all the time.
EDIT: I got less lazy and looked for the card. The one I found, here: http://www.asus.com/us/Graphics_Cards/GTX770DC2OC2GD5/ , doesn't have VGA out. Is this the right card, and you are getting VGA from the DVI-I port? Or is your converter converting FROM hdmi TO vga? Now I'm lost.
EDIT TWO: Yeah, the picture looks like HDMI card output to VGA cable to the monitor. If that's the case, the solution will be easier. Either use HDMI or DisplayPort input to your monitor or, if the monitor doesn't support those, get a digital-to-DVI converter to get a clean signal into your third monitor.
I think I'll stop appending to this and use a new post next time. Please let me know if my guess about your setup is correct, and the monitor model so we can see what inputs it has.