You will get analogue quality, but that should not scare you away. The final quality depends on the cable quality and length, and the quality of the converter itself, as well as the circuitry on the graphics card itself.
https://en.wikipedia.org/wiki/VGA_connector#Cable_quality
With proper cable and all quality components, VGA can drive a very nice and clear image up to 2048×1536@85Hz. That's idel situation, though, so I'd expect that 1080p resolution will work fine in your case. I have used such converters with many older VGA LCD monitors @1080p, and they all provided very nice image quality, I could not tell the difference from DVI. Your mileage may vary, of course.