"some young couple who, on their wedding-journey, at the inn, feel shy in the presence of the waiter"
Signing the Register
Public DomainSigning the Register - Credit: Edmund Blair Leighton

Though this may at first appear to allude to a honeymoon, the practice did not become established in its modern form until the end of the 19th century. Rather, it was customary for newly-weds to repair straight to a good local inn after the service had taken place. As the couple were usually accompanied by the entire wedding party, it is interesting that the governess imagines them to be alone and suggests that the type of union she has in mind is forbidden. 

The simile, in casting the governess and Miles as nervous virgins, also implies a sexual dimension to their confrontation.