I would suggest not. A neurologist will be happy to take your money, but what is he going to add? As 421dog says, you have not definitively had migraines and there is no way of proving or disproving it. Children often have headaches and it is not uncommon to have nausea and vomiting. It is a very brave paediatrician who diagnoses migraine on this basis. A and E is not where you find paediatric neurologists and the fact you stayed for the day means nothing.
I would want to see a common pattern - the same aura, the same pain, the same duration. I would want to see evidence of benefit from specific drugs (although we might not give them to children). and I would want to know there was little benefit from simple analgesics such as paracetamol.
There is a real risk you are asking to be failed. We normally tell people to be totally honest, and that applies here as well, but I do question the reliability of a 6 year old some 26 years later - is the history first hand or are you repeating what your parents said? You had a handful of headaches, they stopped, that is all we can say.