In that case, yes the mother must be a Barred Rock, and the chick must be female.
Whether the chick will lay brown eggs or green, I can't say for sure.
Since the mother laid brown eggs, the chick inherited a not-blue eggshell gene from her. But the father might or might not have any blue egg genes. If he gives a blue egg gene to the chick, the chick will lay eggs that are blue, green, or olive. I'm guessing green, which is a blue egg with brown on the outside, because the mother obviously has the genes to put brown on the outside of the egg, and the chick will probably inherit that.
You can just wait to see what color eggs the chick lays. Or if you are impatient, there is a DNA test. It could be used on the chick (sample of feathers or blood), or they can use the eggshell the chick hatched out of (gooey stuff inside), or testing the rooster could tell what genes he has, which lets you predict what he will produce in future but but may not give a definite answer about this chick. If the rooster has one blue eggshell gene and one not-blue eggshell gene, the chick could have inherited either one. If the rooster has two blue eggshell genes, or two not-blue ones, then you would know exactly what he gave this chick and will give all future chicks.
https://iqbirdtesting.com/blueegg
If I had to make a guess, I would guess green eggs from the chick, but that's a best-guess at present, not a certainty.