Question

What do you mean by private in private quelch?

Answer 1

Private stands for a soldier without a rank.
A soldier who just got into army and is still to earn a rank like a corporal, major etc.

Noctua @_@

Answer 2

Quelch was a mere cadet in his initial days, being trained. He was yet to receive any medals. Hence he was called Private Quelch.
In army the term Private means or refers to a cadet who is still not assigned a rank.

Hope it helps.