Could anyone help me understand this sentence?

Yes, you have it right. Dance partner.

and " piloted Queen Mary through the Royal Quadrille" means leading her to finish dancing Quadrille?
pls help explain this to me

pilot means to steer, so yes, he's steering her through the dance. The quadrille was a dance involving four couples in a square and had set steps each had to perform. If you messed up, you could mess up the rest of the square. It's similar to square dancing today. No one does the quadrille anymore - it was popular in the 18th and 19th centuries.

it's clear to me now. thank you for your excellent explanation, Mame

I think this is also a bit of a play on words. The Queen Mary was a famous luxury cruise liner. The captain of a ship pilots a boat.
It's hard to tell from this short passage, but either the queen was a large woman or the author is praising the Princes timing and dancing skill or it could be both.

The R in Royal should be a small r.