Your question is "how to secure a monitor to a desk", but I believe the actual question should be "How do I prevent my monitor from falling off the desk".
One solution would be to attach it to the desk.
But another solution is to prevent the desk from being kicked.
You may be able to accomplish this by placing things on the floor around the legs, such as a potted plant on the floor.
You could also place the table against a wall, and put the back of a couch against one side.
(No one will walk through the couch!)
Another option is to get a more solid, heavy table that would move very little if accidentally kicked.
You could even reinforce your existing table to add weight to it.
A final option is to place the table against a wall, and mount the monitor to that wall.
It could be just a centimeter above the tabletop
so it is still at an ergonomic height for viewing.
lifehacks.stackexchange.com/users/4198/BrettFromLA
There are 2 epoxy putty brackets, handshaped.
I put plastic wrap on monitor stand during shaping so putty would not leave residue.
Epoxy hardens in minutes.
Then plastic wrap can be removed.
C-clamped for easy moving, but could be screwed down.
https://drive.google.com/open?id=0B3TkWmF8fga3NjhwbnhiaWpDSms
lifehacks.stackexchange.com/users/1818/klausnrooster
Best easy and cheap solution is to find flat type thick ribbon/rope and nail to desk.
We have ships and as you guessed we have trouble with waves.
Our Crew on board found a solution for fixing notebooks,computers even monitors to desks.
It is also easy to remove if you want ; and you can find them from anywhere.
Also we used a professional solution for our general manager maybe you may be interested for that solution too here is the product we used : I hope it helps
lifehacks.stackexchange.com/users/19012/melic
You could use a Spring Clamp on the back of the stand to hold it to the desk:
These are ridiculously cheap to buy and they are useful in all sorts of ways.
If this won't work for some reason, you could basically adhere the stand to the desk by pressing a huge amount of blu-tack into the base and pressing it hard to the desk.
lifehacks.stackexchange.com/users/1965/Lefty
There are some semi-sticky gel pads, if you put them on a table, they stick to the table surface, can be easily removed without a trace by gently pulling for a couple of seconds, but if you try yanking it off quickly, that is really hard.
I saw such things in Japan used for that very purpose (preventing monitors etc.
falling off desks during earthquakes): just put 3~4 of them between the monitor base and the desk surface.
Don't know how they are called in English (or in any other language, for that matter), just google for something like "earthquake gel pads/shock proof pads".
lifehacks.stackexchange.com/users/21176/Headcrab
I want to prevent a computer monitor from falling, if the desk's legs are kicked or anything similar. The desk is relatively light.  I thought of simply drilling a hole through the base of the monitor and the desk and securing it that way. But besides being unsightly, I'm not sure that monitor bases are built for that kind of stress (if the desk is kicked).
Secure monitor to desk (from falling)
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lifehacks.stackexchange.com/questions/15716
lifehacks.stackexchange.com/users/20728/ispiro
There are many different ways to prevent a monitor from falling from a desk. Epoxy putty brackets can be used to hold the monitor in place, as could rope or ribbon. Spring clamps and sticky gel pads could also do the trick. The options are to prevent the desk from being kicked by placing things around it on the floor such as potted plants, place the desk against a wall, use epoxy putty brackets to hold it in place, nail some rope to the desk and tie it with that, use gel pads, use a spring clamp or use lots of blu-tack.
Prevent the desk from being kicked by placing things around it on the floor such as potted plants, or place it against a wall and with a sofa on one side (or mount the monitor on the wall) or reinforce the table to add weight to it. Use epoxy putty brackets to hold it in place. Nail some rope to a desk and use it to secure the monitor in place. Use a spring clamp on the back of the stand or lots of blu-tack. Use the sort of semi-sticky gel pads used to prevent monitors falling over in earthquakes.