In the context of “The Open Window,” a romance isn't a love story; it's any kind of fictional tale showing signs of a vivid imagination. An elaborate ghost story or tall tale would be an example of this. Vera—whose name is ironically often associated with the Latin word for “truth,” veritas—clearly has a very vivid imagination, which is why she's able to come up with such elaborate, well-constructed stories. What's more, she can devise them at short notice—that is to say, she can make them up on the spot, which is no mean achievement.
When Framton arrives at the Sappleton residence, Vera is immediately able to establish that he's a very nervous individual, someone prone to a good scare or two. With that in mind, the mischievous Vera proceeds to conjure up a highly elaborate ghost story involving her uncle and his brothers-in-law expressly designed to frighten the poor, hapless Framton.
It would seem that spinning such stories is something for which Vera has become renowned. That's why we're told that it's a specialty of hers. She's told so many stories like the one she's told Framton that it has become almost second nature to her, which would explain why Framton is so completely taken in by her latest romance.
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.
The word 'romance' isn't a reference to the covert or overt displays of sexually-suggestive affection we are familiar with today. Rather, the term references the 18th/19th Century Romantic movement encompassing literature, art, and music. This movement saw the relaxing of strict rules in poetry, prose, and art forms. The imagination (subjective) was elevated above classicism and realism.
So, the sentence above refers to Vera's gift for making up elaborate and imaginative stories on short notice. She is the type of person who can make up a fantastic story on the spot. Her mischievous sense of humor allows her to concoct a wild tale of morbid terror to horrify the gullible Framton.
After she finishes the gothic tale of disappearance and mysterious deaths, the subjects of her story turn up very much alive. Framton is so spooked that he makes a run for it.
"A most extraordinary man, a Mr. Nuttel,” said Mrs. Sappleton; “could only talk about his illnesses and dashed off without a word of good-bye or apology when you arrived. One would think he had seen a ghost."
In response to Mrs. Sappleton's comment, Vera makes up another fantastic story to explain Framton's sudden departure. Her penchant for imaginative story-telling immediately designates her a romantic who derives great pleasure from her skill.