I was having the same issue. Adding the "?vq=hd1080" unfortunately did not work for my site.
What DID work was:
1. Uploading the video in YouTube as unlisted (originally was using Vimeo)
2. Opening the video link
3. Under settings, changing the quality to 1080 HD.
4.. Copying the link directly from the open video player
I hope this helps someone else as I tried all the other options to no avail!