This statement that Vera is capable of "romance at short notice" is Saki at his satirical best. According to the American College Dictionary, romance is
A tale depicting heroic or marvelous achievements, colorful events or scenes, chivalrous devotion, unusual or supernatural experiences, or other matters that appeal to the imagination.
So, Vera—whose name is ironic since she rarely speaks with veracity, or truthfulness—has now fabricated another tale. This tale is from the real instance of the "tired brown spaniel," using a "colorful" fictional event: Nuttel was hunted by a pack of dogs on the banks of the Ganges. Thus, although this story is set in a faraway land, it is anything but heroic given Nuttel's deeds and experiences.
Vera is capable of blurring the lines between imagination and reality as she entertains guests of Mrs. Sappleton. Her story about the males of the Sappleton family who went hunting and were lost resides between truth and fiction. Later, rather than depicting the guest, Framton Nuttel, as capable of great achievements and "chivalrous devotion," Vera satirically portrays him as a weak and cowardly man who has been pursued by vicious dogs and spent the night in a freshly dug grave to escape injury.
One might say that Vera is bright and imaginative, but one might also say that she is a confirmed liar. The author of the story does not explain that "romance at short notice was her speciality" until after she spontaneously invents a truly wild story to explain why Framton Nuttel fled from the house when he saw the three men approaching with their dog. She tells her story in a single sentence:
"He was once hunted into a cemetery somewhere on the banks of the Ganges by a pack of pariah dogs, and had to spend the night in a newly dug grave with the creatures snarling and grinning and foaming just above him."
She pins the blame on the innocent spaniel and rather perceptively implies uses the Freudian concept that their visitor could have developed a phobia against dogs in general from the encounter with the wild dogs in India. This girl is a very interesting and unusual character who must like to play what are now called "mind games" with people.
It means that Vera can make up a good story in a moment's notice, or very quickly. She has just demonstrated this quality by telling Nuttel the story of "The Open Window" and again when she is asked why Nuttel left so quickly. She lies and makes up stories that she tells to other people just to amuse herself. She does not seem to care about the consequences of her fictional work on others. Poor Nuttel is driven insane but Vera simply goes on and makes up another story.
"Romance at short notice was her specialty" is Saki's way of saying that Vera was good at making up stories quickly. Her tall tales, such as what she told Framton Nuttel, are what she excels at. She can make even a boring countryside and a boring country house romantic and exciting, if only with her stories. It is her poise that allows her to be believable even when she is making up totally false tales.
In that particular story, "romance" refers to the genre of imaginative literature, not romance as in "love." The girl in the story (I think I her name is Vera) is adept at inventing stories (i.e. lying) on the spot. Thus, her story about the men getting lost in the bog during their hunting expedition, etc., as an explanation about why the window is open. Simply put, she's good at making up stories.
In this context Saki uses the word "Romance" to mean made-up stories of a wild or improbable nature. This meaning of the word is older than the modern association with falling in love(although this usage was also known in Saki's time.) It used to be quite common to accuse someone of "romancing" if they told you a tall or improbable story.
She took advantage of Mt Nuttel ignorance to deceive and make fun of him.She is careless what is the implication of her action .It may further detiorate his medical case.