...small square of grass? Where does this term come from? Maybe I have misremembered and it's called something else entirely? An idiot of turf? A git of grass?

sod or turf. Just saying (I am gonna lay "turf") is equal to saying (I am gonna lay "sod") There is an understanding there. One term could be English(sod) and turf may have come from AstroTurf terminology.

Much easier than saying, I am going to lay pre grown lawn.

Are you in UK ? Sounds British *shrug* Always just went with "sod" . Sod is sold by the yard (9sq ft.) Before mechanical harvesters , it was cut in 3'x 3' squares . Now it is 6' x 1 1/2' . Saw Zoysia , in Florida , cut 3' x3' , but was when the sod farm was closed , so didn't stop to ask about the details . ( In the biz at the time ,in NY, so not as wierd as it sounds )