I am 17 years old and my girlfriend is 16 years old. We have a 2 month old son. My girlfriend is very immature and she starts arguments over nothing and when we argue she says she is going to take our son from me. What can I do to assure the fact that I will have my son. I'm afraid she will take him and run to North Carolina with her family. What do I do??

Do you have parents or family you can confide in? I'd lean on them for advice & support. Do you live with your girlfriend? Either way, that threat would have haunted me as a young father too.
To answer your main question is that you have to get the ball rolling with the courts immediately. Make sure your paternity is confirmed (a copy of your son's birth certificate with your name on it will do, unless she tries contends something different).
First of all you should get an attorney (but don't inform you girlfriend since you seem not to totally trust her). If you can't afford a lawyer depending on where you live the county courthouse will possibly assist with either an attorney or advice. But expect many, MANY phone calls & the run-around, tons of visits to courthouse. Ask every question you don't know the answer to & don't give up!
In some instances you may either get joint custody or if the judge seems she is unfit you may get awarded temporary full custody until the situation gets fully evaluated. In any event, what the judge will have to go by is the law & it is simply put: "In the best interest of the minor child" & seeing that the two of you are young the judge may consider how the baby will be provided for: do either of you work, if not who supports you & the baby, do you live together or do you each live with your own parents? Im sure many other factors can come up as well.
Even if she ends up the custodial parent (which parent the child lives with), you can at least retain your father's rights & if your son does live with her that means child support, but that also means she cannot leave the state with your son without your consent or consent from the courts. Even if you aren't paying child support she CANNOT withhold visitation rights. The two are entirely separate cases that one does not affect the other. Best thing is to pay, your son will need it.
One thing that happened to me when my daughter was around 9 months, at least in Illinois, the court only awarded limited visitation due to our baby being so young & the fact that she already lived with mom. My lawyer said it was typical & I, after some time, had to go back to request additional joint custody & she remained custodial parent, but we have managed to work together for the most part.
Whatever the outcome it's nice to see a young man parent a child be so intent. Keep it up & you will be a better man for it... so will your son someday.
My daughter is your age with a child & I Thank God the 2 of them are together & doing well. I wish you all the best with your son & your relationship . Who knows, maybe she will mature. Women, after giving birth can be going through a lot, so if you love her try to make it work. Give her support, be patient, show her you want to be a part of her life & to be a great father. The best for the three of you is to work through the hard times, it will make the good times that much more rewarding. I wish I tried so much harder to work things out with their mom when my kids were younger. You won't regret being there.
Good luck

Without knowing more ? Are you living together, if so where, with parents, on own.

Are you listed as the father on the birth certificate. And of course I did not see the word married anywhere either.

If you are no living together, there should be a child custody agreement, in it, would be what rights you have to visit the child, conditions on her leaving the state and what support you have to pay each month.

Something else you should be looking at is that if the child is only 2 months old there may be other factors coming into play. 1) sleep deprivation ( those nightly feedings can be tough and sour a relationship ) 2) post partum depression ( it comes after having a baby ) can also be showing up.

The courts are your only remedy for ensuring your time with your child. Also you will need to eastablish child support since your not married. It sounds odd but its true.