I use to eat at a Japanese restaurant in Florida, they had a steak with a brown sauce on it that tasted somewhat sweet but grilled, can you suggest what it may be?

That would be teriyaki and yes Hannah Steakhouse does have it they do a great meal

I abstain from meat therefore i am not familiar with the sauce.

Ooooh the teriyaki sauce for steak is the best! Hana has that and a orange sauce that is also phenomenal. If you're in the area I highly recommend it.