The short answer is maybe, but probably not much. In large scale studies of the birth control pill, patch, and ring, it was demonstrated that some people did experience weight gain but almost never greater than 2-4 pounds that could be linked to their birth control.
It is likely that there are other factors that contributed to much of the weight gain in these studies, because it happened to people on and off birth control. However, every body is different and we still don't fully understand the role of these hormones in weight gain - so if you are experiencing abnormal weight gain let your HeyDoctor medical team know or discuss it with your in-person doctor.
This is important because not only are there excellent non-hormonal options like the copper IUD that may be better for you, but abrupt weight gain could also be the sign of other problems like thyroid disorder.