betraying the slightest suggestion of a smile.
That means simply that the captain was smiling very slightly, probably without intending to. It shows that he thinks Rostov is naive to believe that these are his quarters.
Merriam-Webster "betray"
4a : to reveal unintentionally
betray one's true feelings
A suggestion here is an indication or appearance.
The word slightest means that it is a very small indication, so it's a very small smile.
The OP has a quotation from "A Gentleman in Moscow" by Amor Towles.