Does grass turn into hay

No. Grass only is "turned into" hay when it is cut and collected for the sole purpose of feeding animals i.e., livestock. Grass will be grass no matter if it's alive and green, or brown and dead.

Besides, FYI, grass hay is cut when grass is green, not when it has reached full maturity and gone into dormancy (i.e., turned brown and "died"). If grass is cut at this stage it contains very little nutritional value for livestock, and is otherwise considered the same value as straw.