Using generic methods as an alternative to overloading?

Both ideas are bad. Your ObjectA and ObjectB classes should implement a common interface that defines getId() and getName() methods.

Then you can get away with a single method:

public String getInfo(Interface o){
    if(o.getId()!=1) return o.getName();
    return "";
}

Your case is a good candidate for Programming to Interface. Create an interface and move common methods.

public String getInfo(InterfaceAB in){
    if(in.getId()!=1) return in.getName();
    return "";
}