Is there a way to link someone to a YouTube Video in HD 1080p quality?

No, this is not working. And it's not just for you, in case you spent the last hour trying to find an answer for having your embeded videos open in HD.

Question: Oh, but how do you know this is not working anymore and there is no other alternative to make embeded videos open in a different quality?

Answer: Just went to Google's official documentation regarding Youtube's player parameters and there is not a single parameter that allows you to change its quality.

Also, hd=1 doesn't work either. More info here.

Apparently Youtube analyses the width and height of the user's window (or iframe) and automatically sets the quality based on this.

UPDATE:

As of 10 of April of 2018 it still doesn't work (see my comment on the accepted answer for more details).

What I can see from comments is that it MAY work sometimes, but some others it doesn't. The accepted answer states that "it measures the network speed and the screen and player sizes". So, by that, we can understand that I CANNOT force HD as YouTube will still do whatever it wants in case of low network speed/screen resolution. From my perspective everyone saying it works just have false positives on their hands and on the occasion they tested it worked for some random reason not related to the vq parameter. If it was a valid parameter, Google would document it somewhere, and vq isn't documented anywhere.

Yes there is:

https://www.youtube.com/embed/kObNpTFPV5c?vq=hd1440
https://www.youtube.com/embed/kObNpTFPV5c?vq=hd1080
etc...

Options are:

Code for 1440: vq=hd1440
Code for 1080: vq=hd1080
Code for 720: vq=hd720
Code for 480p: vq=large
Code for 360p: vq=medium
Code for 240p: vq=small

UPDATE
As of 10 of April 2018, this code still works.
Some users reported "not working", if it doesn't work for you, please read below:

From what I've learned, the problem is related with network speed and or screen size.
When YT player starts, it collects the network speed, screen and player sizes, among other information, if the connection is slow or the screen/player size smaller than the quality requested(vq=), a lower quality video is displayed despite the option selected on vq=.

Also make sure you read the comments below.