I would say that righteousness as it applies to God would mean that everything He does is right.
He makes no mistakes. He doesn't "miss" His own standard of perfection.
We, on the other hand, do nothing right of ourselves. Even when, on the surface, our efforts may appear to be "right", there is always a hidden motive behind what we do, whether we are aware of it or not.
In the creation, God displayed some of His hidden attributes; His power and his divine purpose. But, He had not revealed His righteousness in fullness until Jesus came.
Everything Jesus said, thought, did, and did not do, was "right". He was the Lamb without blemish.
Only in the gospel as presented by Paul is God's righteousness revealed.
Sending His Son, in the likeness of sin's flesh, was the purest of right things to do.......and, sending His Son to the cross was even yet more pure and right still. It has "righted" all wrongs.
When God does something, the glory of God emmanates out of the act. Righteousness radiates a glory unequaled by any creation, or any creature.
Most amazing is that He has chosen to share His glory with us. First, however, He shares His righteousness when He justifies us in His blood.
The weight of our current outward body and the pressures that are brought to bear cannot be compared with the glory that we will share when we are in His presence.
For I am persuaded, that neither death, nor life, nor angels, nor principalities, nor powers, nor things present, nor things to come, Nor height, nor depth, nor any other creature, shall be able to separate us from the love of God, which is in Christ Jesus our Lord. Romans 8:38,39