What do you call a green-land right behind a sea-shore?

Not a single word, but you could call it strand vegetation. Also read this book about 'strand vegetation'.

There's also 'hinterland' which means the land behind the coast or the banks of a river, or an area of a country that is far away from cities, but I don't think it's what you're looking for.

In oceanography and coastal ecology I know no single word but the phrase coastal fringe is often used of the vegetated land immediately beyond the reach of the tides, or indeed just in reach - as in the case of mangroves. It has the advantage that fringe suggests the disorder of a natural vegetation.

To give just one example: "Ecological Effects of Nutrient Enrichment on a Coastal Fringe Mangrove Forest Mangrove Forest"

(https://core.ac.uk/reader/56698320)

Beachfront flora

flora
The plants of a particular region, habitat, or geological period.

Wikipedia

Beachfront flora plays a major role in stabilizing the foredunes and preventing beach head erosion and inland movement of dunes. If flora with network root systems (creepers, grasses, and palms) are able to become established, they provide an effective coastal defense as they trap sand particles and rainwater and enrich the surface layer of the dunes, allowing other plant species to become established.