Can a Christian have a relationship with a non-Christian?

I think you can.
I've never heard or any rule stating that you can't - Hey look at me, I stopped following religion but I'm with a guy who still goes to church :')
It they both believe in God, they will get along better than if the non-Christian was an athiest.

For me I think the answer to that is definitely a yes if you really love her then go you can even try to influence her to be a christian but if she don't agree the you should just respect each other belief

You surely want to avoid becoming “unevenly yoked” with someone who could take you away from God. (2 Cor. 6:14)