The sound starts when I am moving at about 20 mph or more. It is not when I am braking but it does sound a lot like grinding brakes. If anyone knows what this might be advice would be greatly appreciated!

Were the caliper pins cleaned and greased? Was the brake fluid completely changed? If not, the caliper pistons may not be retracting enough, causing the brake pads to remain in contact with the rotors.

If the problem has gone too far, the calipers may need replacing. This usually happens when the brake fluid is not changed every 2 to 3 years.

This link may help:

https://www.askmehelpdesk.com/cars-trucks/faq-how-troubleshoot-repair-maintain-hondas-selected-other-vehicles-46563-9.html#post1618425

I'm not sure about the pins. I took it to a shop and of course the problem didn't start until I had already left and the shop had closed. The brakes are working perfectly, by the way. I'm debating if I should drive it to work tomorrow.

Also... I have only had the car for a few weeks and don't know the history.

Attention to detail makes a big difference in brake jobs. It's important to use top quality pads, rotors, brake fluid, and silicone grease. Use the link I gave you to pass judgment on what they did or didn't do.

Thank you. I will do that.

You may also want to read up on Lateral Runout (LRO) problems. It could be the rotors were turned to the axis on the lathe and not the axis on the spindle. The difference is making itself heard as the pads rub against the rotor. The best way to correct this problem is to use a Pro-Cut machine, which is the only method endorsed by most car manufacturers.

Could be as simple as a dust shield or backing plate rubbing on the inner edge of the rotor.
Sometimes these are grabbed and pulled but not intensionally when cranking the front wheels left or right by hand while performing the brake job.
It makes a horrible grinding noise as you describe but is really not a safety issue.