Regarding the question
I reset both of the monitors to their factory defaults, but still, the issue persists.
This does not help at all. The basic configuration is not on the monitor but in the graphics card that sends the signal to them.
For web could be enough, but for photography or print, you need special hardware. There are some brands, for example, this one: http://www.xrite.com/colormunki-display
The basic calibration I mention is so you know, at least, your colors are not too crazy.
I'm wondering if some of my clients might dislike my designs because of this. What should I do?
TALK to them. "Hey, the color for this can vary from monitor to monitor, this depends if your monitor has some basic calibration"
How can I make sure that what I design is what my client sees?
If exactitude is needed, either deliver a calibrated print sample in case of printed material or go with the client and calibrate their monitor or show it on a device of your own... calibrated.
Did you notice the word calibrated?
You need to make a simple basic calibration on your 3 devices. (It can be hard to modify the basic brightness and contrast of some led monitors, but try to find how to specifically adjust them)
A basic tutorial
Many new monitors are limited on the options you can adjust. Some may be present or not.
1) Monitor temperature - color
On the monitor's menu look for an option temperature/color. Some monitors define this as temperature or as the specification. Choose:
Do not use, warm, cool, blueish. Sometimes you can use 5500°K but the recommended is D65.
If your monitor can handle this normally a brighter option is better. "Brightness" controls the darker points of the image.
Here is a test pattern. You need to barely see the No.1 using a dark background. Open the image in a new window.
But as this is too easy to move in some devices is the hardest to maintain well defined.
The most crucial parts are the middle tones, which are controlled by the gamma function.
This is controlled by the graphics card. Look in your control panel or in windows right click on the screen. There is a chance you have some kind of application to modify this. Something like this:
You need a "standard target" to compare. You can adjust all the RGB components together, but I recommend setting them one by one.
The objective with this patterns is to modify the gamma settings so the 2.2 number blends with the background. Higher numbers will look brighter and lower numbers darker.
Open this images in a new window, at 100% zoom:
Almost all modern computers have this panels, but in the case, you do not have it you can use this application on windows. http://www.quickgamma.de/indexen.html
Vincent posted one comment that wants to be addressed:
Why should I do it?
Here is a joke:
One man is driving a car and he is hearing the news. "One drunk man is driving on the opposite way on the highway!"
He says... One? There are a lot of them!
Well, you do not want to be the drunk man.
If your calibration is too magenta, and you shift the image to greens, the image is green, not really balanced.