How do I make the method return type generic?

You could define callFriend this way:

public <T extends Animal> T callFriend(String name, Class<T> type) {
    return type.cast(friends.get(name));
}

Then call it as such:

jerry.callFriend("spike", Dog.class).bark();
jerry.callFriend("quacker", Duck.class).quack();

This code has the benefit of not generating any compiler warnings. Of course this is really just an updated version of casting from the pre-generic days and doesn't add any additional safety.

No. The compiler can't know what type jerry.callFriend("spike") would return. Also, your implementation just hides the cast in the method without any additional type safety. Consider this:

jerry.addFriend("quaker", new Duck());
jerry.callFriend("quaker", /* unused */ new Dog()); // dies with illegal cast

In this specific case, creating an abstract talk() method and overriding it appropriately in the subclasses would serve you much better:

Mouse jerry = new Mouse();
jerry.addFriend("spike", new Dog());
jerry.addFriend("quacker", new Duck());

jerry.callFriend("spike").talk();
jerry.callFriend("quacker").talk();

You could implement it like this:

@SuppressWarnings("unchecked")
public <T extends Animal> T callFriend(String name) {
    return (T)friends.get(name);
}

(Yes, this is legal code; see Java Generics: Generic type defined as return type only.)

The return type will be inferred from the caller. However, note the @SuppressWarnings annotation: that tells you that this code isn't typesafe. You have to verify it yourself, or you could get ClassCastExceptions at runtime.

Unfortunately, the way you're using it (without assigning the return value to a temporary variable), the only way to make the compiler happy is to call it like this:

jerry.<Dog>callFriend("spike").bark();

While this may be a little nicer than casting, you are probably better off giving the Animal class an abstract talk() method, as David Schmitt said.