No, that would not be the problem. Lightroom can manage images on any hard drive that is connected to your computer. In all probability your monitor is set so that it is too saturated and too bright. Your monitor probably needs to be calibrated. The best way to do that is to use a hardware calibration device. I have adjusted my monitor using the Windows tools and the adjustments on my monitor. The real "pros" will tell you that is not a good way to do it. And I know that. But it has worked for me. I adjusted my monitor until the monitor looked like one of the images or prints. And then adjusted accordingly in Lightroom.
Have you checked the profile used when exporting? For anything web based I use the sRGB profile; only if I am printing I use a different custom profile.