How can I seek forgiveness for believing men and women for the sake of reward, excluding specific individuals for whom I do not wish to seek forgiveness?

A Muslim is recommended to seek forgiveness for believing men and women (Al-Mu'minīn wal-Mu'mināt), and this is a characteristic of the pious. One receives reward if they intend to seek forgiveness for them. A Muslim is a brother to another Muslim, and one must desire good for them. Forgiving and overlooking others are causes for Allah's forgiveness and pardon. As for seeking forgiveness for Muslims in general, as long as you did not intend in your heart to seek forgiveness for a specific person who wronged you, then that does not mean you have forgiven them, for actions are judged by intentions.