Since I have a family history of fibroids, am I more likely to I get a uterine growth during my pregnancy?

Yes. Yes, but. Your genes are merely your potential. If you show your genes different nutrition, exercise, thoughts, beliefs, and ways of dealing with stress, you do not have to manifest fibroids. That being said, fibroids that show up on your ultrasound during pregnancy were probably already there prior to conception.
Yes. Fibroids, which are benign muscle tumors in the walls of the uterus, can grow during pregnancy from increases in estrogen. This doesn't mean they will or that it will create a problem. Complications due to fibroids include miscarriages, premature labor, excess bleeding at delivery. Your OB will be individualize your care based on location and size of fiborids, so good to evaluate beforehand.