Is there a specific word for "developing a liking towards someone after someone else says they are nice"?

A begins to like C by extension.

B's good opinion of C is contagious.

A slight variant of your situation: A and B are friends, and B is a fan of C. Let's say B talks up C in conversations with A. Then we can say that B turns A onto C.

And now I'll invent something for this purpose. If A and B are friends, and B and C are friends, then A and C are likely to become friends by the transitive law of friendship.

If OP is still interested in this and is particularly interested in one or another of my suggestions, or is confused about anything, I'd be happy to provide documentation.