Quai hai?

I don't know the meaning, but I can give you the proper spelling. Oddly enough, I'm in the process of rereading the books right now! It's "Qui hi."

"If ah irritate you, jes think how ah irritate mahself."

qui-hi
n. Anglo-Indian call for a servant.

“Qui hi, or Kooee Hye, is the Bengalee phrase meaning Who’s there. As no bells are used in the houses of the gentry in India, the servants sit at the doors of the dining or drawing room, or study, and are called … by the phrase Qui hi, to which they instantly reply by appearing. Hence all the English in Bengal are called by the Europeans of other parts of India Qui his.”

— Joachim Stocqueler, Familiar History of British India (1859)
http://blogs.princeton.edu/graphicarts/2012/03/adventures_of_qui_hi.html

There were several times it is used in the episodes where it makes no sense, meaning "servant." (ie, complimenting Lucia on her lobster dish, saying goodbye in the street, etc.)

So it's a name for All English people, as well as servants.
Seems like they use it as a catch-all phrase, too.

Qui hai, Hindi for "Who is there?" of "Is anyone there?" (It's in Chambers, among other dictionaries)
Since it's Indo-European, it's basically the same as in French and Spanish (French 'qui', Spanish 'hay').
The major does use it inappropriately, and I take it to imply that he has less Hindi than Lucia and Georgie have Italian.