No, in my opinion it's not weird. 'I'm starting to like you' is what it means, though you are probably not going to hear it expressed in this way too often,

"you're growing on me" = "you're starting to like me" If you felt a likeness on someone (whom you don't like at first ) then you can say, "I'm growing on you". "Growing" defines the development of your feelings toward someone. good luck! ~romulus

If someone or something 'grows on' you, the assumption is that you didn't really like him/her/it to begin with, but you are starting to. i.e. "I didn't like sushi before, but it's growing on me."