You can use call() or apply() like this:
this.getContactName(id, (error, contactName) => {
  if (error) return callback.call(this, error);
  // want to access "this" here e.g., 
  // this.indexScore = 1
  return callback.call(this, contactName);
});
Or with apply()
this.getContactName(id, (error, contactName) => {
  if (error) return callback.apply(this, [ error ]);
  // want to access "this" here e.g., 
  // this.indexScore = 1
  return callback.apply(this, [ contactName ]);
});
Both methods bind the first argument as this value. The difference is, that apply() has an array of function arguments as a second parameter whereas call() just has one more argument than the initial function call (the first one is the functions's this value). See this answer for more information.