Well Eye color has different forms in its gene, known as alleles. Eye color also has a dominance rule, which in your case it's between the two alleles Brown and Blue. Humans can have 2 alleles for each gene only. As an example for this gene we will go with the letter A, thus A means brown and a means Blue.
Having at least 1 A in the Eye genes makes the eyes become Brown. So the genes AA and Aa are brown and aa is blue.
Now you mentioned your parents have brown eyes and you have blue, so they both must be Aa. Because you get one of these alleles from your father and one from your mother. So you have got a from both of them which makes your Eye gene aa.
Now if that lady of yours has Brown eyes, she could either be AA or Aa, if she is AA then there is absolutely no chance the child that would come from you two (Don't be optimistic that there will even be a child) would have blue eyes. If she is Aa there is a 25% chance that the child will get blue eyes.
It's quite more complicated but I still think it's hard enough for you to understand.
))
