My hyperactive border collie/springer spaniel has been eating a lot of grass the past couple of days. She normally jumps up when people come to the door, she normally runs around like crazy. But today she is like a statue, she's very still and seems to want me to be right next to her all the time. I noticed it right away. Can dogs get stomach bugs? Is there anything I can give her to make her feel better?

Pica ( eat non-food items) is actually very common in dogs but is she thowing up too? The quiet behavior tells me she is not feeling well. Is she eating her regular food and drinking water, what about bowel movements?

I don't know if you could actually call what dogs get stomach "bugs" but yes, they can suffer from GI issues. Sometimes when they eat grass it results from a natural instinct to purge themselves of some offending intestinal issue. In other words there stomach doesn't feel right.
What is going on with your dog, I can't say, I could only guess. So taking her in for a check especially if you feel that she is uncomfortable is highly recommended, your vet may want to check a fecal sample to check for causes that could range from parasites to bacterial overgrowth in the intestines. If she is a dog that has a habit of getting into things or chewing up toys, clothing, shoes, etc. these present another concern of chemical ingestion to obstruction.
Mind you I am brainstorming here, some of this may not apply to your dog, but I would advise a check over, if she is not her normal bouncy self.
Good Luck I hope she is feeling better soon.