There doesn’t seem to be a set “can” and “can’t” eat list out there, the ones that there are some things are correct and some things are not.
However, generally do not give them fruits of veggies that have a high moisture content as this can cause the runs, and do not give them any fruits or veggies that are acidic (ie: oranges, tomatoes, etc). Also no avocado, raw potato, etc.
I’ve heard not to give them lettuce, cabbage, or broccoli however some people say it’s ok. Those types of veggies apparently can cause wet tail, according to some breeders.
Also no peanuts, cheese, or anything that has a high salt or sugar content.
Generally you can give them small portions of apples, pears, carrot etc. No need to cut them up small – gerbils can gnaw and chew. My gerbils like a slice of apple or pear occasionally but I don’t give them fruits or veggies often. Too much of any kind gives them the runs. Use once or twice a week as a general rule – no more then that.
Other treats unrelated to fruits and veggies – pick out the sunflower and pumpkin seeds from their food mix and save those as treats. Popped unsalted/unbuttered popcorn is another favorite (not the microwavable variety). Rice Krispies, Cheerios, plain bread can also be used as treats.