Dear Tiffany,
First, there might be a miscommunication going on between what your mom was told or what she told you. Many women when pregnant are tested to see if they are carriers of the CF gene. She might have been told she was a carrier and misunderstood that, thinking she was diagnosed with CF. Or maybe you misunderstood what she said. So the first question to find out is whether she was told she was a carrier or she had some specific test for CF.
The reason your doctor likely isn't taking the idea seriously is if someone is diagnosed with CF, they are treated. It's not just a "oh, I have this disease" kinda thing. It well could be she has such a mild case she has had issues but have been fine without treatment, but a doctor is going to find that hard to believe because most doctors only know of the "typical" CF cases. So, in order to get the doctor (this one or another) to take you concern seriously, you'll need to get your mom to help. If she was diagnosed with CF (and not merely told she was a carrier), then you need her to get her medical records, find out how she was diagnosed (genetics, sweat test), where and then get back to an accredited CF clinic for another "diagnosis" to give your doctor; and at the same time in getting that diagnosis, she could work with that CF Center to discuss how to get you tested. Note: Just because your mom has CF doesn't mean you have it. Your father would need to also be a carriers, and then it is a 50/50 shot if she has CF.
If she was merely a carrier, then you want to know that too and the specific mutation because even if she doesn't have CF, but is only a carrier, you would then have a 25% of having it IF your father is a carrier. You doctor may be more open to testing if you are able to clarify that she is a carrier and that you do/don't know re your father, but that there are many cases of late diagnosis. For this later point, you'll want something to show the doctor or refer him to.
But first, please follow up on the above and then repost and we'll all try to help you get a game plan.