In Romans, the righteousness of God is God's own righteous action revealed in the gospel. God is faithful and just: he exposes sin, judges evil truthfully, and provides salvation through Jesus Christ for those who believe.
Romans 1:16-17 introduces God's righteousness as the gospel's revelation. Romans 3:21-26 then explains how that righteousness is shown apart from the law: God justifies by grace through Christ while remaining righteous himself.
This phrase holds together two truths Romans refuses to separate: God does not ignore sin, and God saves sinners by grace. The gospel reveals both God's justice and his mercy.
A careful reading should not reduce the phrase to only a moral quality or only a status given to believers. In Romans, it is tied to God's saving and judging action in Christ.
Original synthesis from Romans 1 and Romans 3.
17For therein is revealed a righteousness of God from faith to faith. As it is written, "But the righteous shall live by faith."
21But now apart from the law, a righteousness of God has been revealed, being testified by the law and the prophets;
26for the showing of his righteousness at this present time; that he might himself be just, and the justifier of him who has faith in Jesus.