I'm certainly aware that the term became a sort of code for "nice, white suburbs," so I consider whether the use is in that context or in the historical context of early planned suburbs from the latter half of the 19thC to, say, the 1930s.
Once at the edge of cities, these "leafy suburbs" in the U.S. grew out of the City Beautiful Movement, the designs of Frederick Law Olmsted and his sons, from British "Garden Suburbs" like Bedford Park at the then edge of London (from 1875 to about 1905) and early 19th suburban movements on both sides of the Atlantic, the railroad towns of Philadelphia's Mainline, etc.
They are now "close-in suburbs" with some age, sometimes with a grid plan but often with curvilinear streets to break from the urban grid and mark a solidly residential area with middle class, upper middle class, and sometimes houses of elite classes: mostly free-standing houses set back from the street with deep rear gardens, the houses were usually built in a variety of styles of a period of one- to three-decades that the area developed, and yes, they are leafy and green, with sycamores and elms and oaks and maples (common street names for these neighborhoods), and shrubs and hedges, and a patch of front lawn. Larger planned suburbs like Roland Park in Baltimore (in part by the Olmsted Brothers) and SHaker Heights in Cleveland had separate provision for shops for everyday needs like grocers and butchers and bakeries, usually tucked away at an edge of the suburb or separated from the streets of single family houses. Sometimes there are apartment buildings of the same period incorporated into the schemes but that's a bit unusual.
To me they are the good suburbs, because they are handsome historic houses, in a beautiful setting with canopies of mature trees, in or immediately adjacent to the city center. Even on "small" lots (which I prefer but many Americans do not) they make for very attractive neighborhoods and provide a good measure of privacy, the greenery itself often contributing in no small measure.
The photo is of Schenley Farms, a small suburban corner of the Oakland neighborhood in Pittsburgh, Pennsylvania, begun about 1905 as a "model city" incorporating elements of CIty Beautiful Movement planning. It has free-standing houses, terraced houses, and I think some apartment buildings in a range of Tudor Revival and Colonial Revival and Eclectic Period styles. The overall planning of the neighborhood elevates the appeal of individual elements which range considerably in forms, size and, decorative detailing and were meant to appea to people of different budgets and needs. Very different from modern "cookie-cutter suburban houses" where the choice is of three sizes of houses in three different sectors of a development, choice for early buyers of a snout garage door on the left or right, and scarcely a tree in sight.
It's possible to have a new built "leafy suburb" but the implication is of a place established for many years and with a richer variety of architecture than in a modern development. So leafy can be "code" or just descriptive of a physical type of neighborhood.