It's listed in a number of spurious quotes wrongly attributed to Lincoln here and here.

If you conduct a word search in the collected works of Lincoln, the word "adversity" never appears.

The quote also does not appear in the Recollected Words of Abraham Lincoln.

The author of the quote was in fact Bob Ingersoll, a 19th-century jurist and staunch Republican who was famous for his defense of agnosticism. The quotation itself refers to Abraham Lincoln, but was written by Ingersoll in this 1884 essay:

Nothing discloses real character like the use of power. It is easy for the weak to be gentle. Most people can bear adversity. But if you wish to know what a man really is, give him power. This is the supreme test. It is the glory of Lincoln that, having almost absolute power, he never abused it, except upon the side of mercy.