Yes they do, the latest of which is shown in the below image: the Adidas X 16.3. The Adidas X 16.3 features the 'Chaos' outsole: a moulded cleat configuration that Adidas states: "provides explosive acceleration on turf surfaces." Turf shoes tend to use a cleat, rather than stud, configuration: where moulded projections are designed to prevent the user losing their footing and ideally to provide optimal grip and propulsion.
Adidas' history is intertwined with that of 'turf shoes/boots': one of it's early shoes, the Adidas Samba, released in the 1950's, was a shoe designed for footballers to train with on icy pitches. The Samba is Adidas' second highest selling shoe (in units sold), behind the Stan Smith, and is still in production to this day. Of course, the Samba is a heritage product, and Adidas have made numerous training/turf shoes in the intervening years. Advanced manufacturing and design has enabled Adidas to produce outersole's which are far superior - when compared to the Samba - at gripping artificial and natural turf.