Following the Ngrams hint, I eventually located an illustrated 1903 edition of _Tales from Maria Edgeworth_. Page 91, near the beginning of the story "Simple Susan", contains an illustration by Hugh Thomson matching the description of a "farm half hidden by trees" in _The New Treasure Seekers_ :
![A man on a horse wearing a black hat is talking to a young woman wearing a polka-dot dress standing on some steps under a tree. In the background a cottage is half-screened by trees. A caption reads, "The gentleman asked her how many miles it was".](
This story had been in publication since 1796, though the illustration is newer. Given how well it matches the quotation, I'm willing to guess that this was what Nesbit was referring to. Thanks to Rand al'Thor for the Ngrams search hint and to Tsundoku for such a detailed answer!