While there may be a FiOS problem, I can tell you that I have used HDMI to my Sony LCD HD TV for 3 1/2 years without any problems. So it can work.
Now, if HDMI doesn't work for you, why aren't you using component cables (or are you)? They will provide HD content to your TV, just like an HDMI cable will. The picture may or may not be as good as using HDMI, some people say HDMI is better, some say component is better. And if you are already using component cables, you most certainly should be able to get HD content in HD.
So, lets start with:
1) What Verizon STB do you have?
2) How do you have the Verizon STB connected to your TV since you can't use HDMI?
3) What level if the IMG do you have now? If you don't know, Menu ==> Settings ==> System Information
4) What FiOS TV package are you subscribed to?
5) Where are you located?
Once we know this info, hopefully we can give you some additional advice that will get you the HD picture you should already have.
Verizon FiOS TV, Internet, and phone
IMG 1.6.0, Build 06.89
Keller, TX 76248