When God's righteousness is mentioned in the gospel, it is God's action of declaring righteous the unrighteous sinner who has faith in Jesus Christ. The righteousness by which a person is justified (declared righteous) is not his own but that of another, Christ.
Martin LutherO, this faith is a living, busy, active, powerful thing! It is impossible that it should not be ceaselessly doing that which is good. It does not even ask whether good works should be done; but before the question can be asked, it has done them, and it is constantly engaged in doing them. But he who does not do such works, is a man without faith. He gropes and casts about him to find faith and good works, not knowing what either of them is, and yet prattles and idly multiplies words about faith and good works.
Martin LutherTo be convinced in our hearts that we have forgiveness of sins and peace with God by grace alone is the hardest thing.
Martin Luther