There is a provision in the Constitution for the President to propose bills to Congress. From Article II, Section 3 (emphasis mine):
He shall from time to time give to the Congress Information of the State of the Union, and recommend to their Consideration such Measures as he shall judge necessary and expedient;
While every President has had the ability to propose bills to Congress, I can't find any specific examples of either Washington or Adams doing so. The earliest example that I found was from 1806, when Thomas Jefferson called upon Congress to criminalize the international slave trade. The bill passed and Jefferson signed it into law in 1807, but it didn't take effect until January 1, 1808 (the earliest that the Constitution allowed the slave trade to be banned).
There are other notable bills proposed by Presidents throughout U.S. history. For example, in 1909 William Howard Taft proposed a federal income tax that became the 16th Constitutional Amendment. In 1933 Franklin Delano Roosevelt started proposing a number of bills as a part of his New Deal. According to Wikipedia, "from March 9 to June 16, 1933, he sent Congress a record number of bills, all of which passed easily." So, while every President has had the ability, FDR was the one that made its use more common.