You might get a good reception but GP more likely to look for evidence in the form of symptoms. Going private you could get an MRI and an interpretation by a consultant neuroradiologist.
As @GCCK indicates your GP can’t do much in the absence of any symptoms. Quite often, including in my case and way before MS diagnosis I experienced a range of odd but not troubling symptoms and didn’t take any notice of them e.g I used to be a hill walker and suddenly started experiencing vertigo in the most innocuous looking hill/ mountain sides. Thought that was odd but didn’t do anything. I also had some strange tingling on one side of my face. If you have experienced any unexplained ( but real)symptoms then no harm in telling your GP about this and your family history of MS ( probably worth telling him/ her anyway just so they have it on record).
So far as I know there is no test to determine your likelihood of developing MS.
As I understand it, way before any symptoms appear there can be sort of hidden / unnoticed symptoms and MS lesions in your brain. If you can then getting a private MRI is one option for you and should show any current but symptomless lesions. It’s a real sod though - so far as I know, the absence of any current lesions isn’t a guarantee that you won’t get any in the future.
P.S. I was diagnosed in my late 50s but other than my aunt (Dad’s sister) there seems to no family history - my Dad (died in his late middle 90s) didn’t develop it nor my two sisters ( in their late middle 60s). I can’t remember the figures for sure, but I think having relatives with MS increases your chances by less than 5% ( worth checking that figure)
My brother felt like he was drulling down one side of his mouth. Got the MRI scan and was 4 lesions on his Brain which were signs. Just had the spinal fluid taken too. Not sure if it’s paranoia or not. Get really itchy especially round my legs also get all I can describe it as a sudden woosh of dizziness dursnt last long.