(covers information from several alternate timelines)
Fosterage was the act of raising an offspring that was not one's own as if it was one's own, but unlike adoption, the actual parents were generally acknowledged as the rightful parents. The term surrogate was applied in post-partum (beyond being a surrogate parent) or towards a young adult (where one may seek out or require a father figure).
Sergey and Helena Rozhenko were the foster parents of a Klingon named Worf, whom they raised as their own son. (TNG: "Heart of Glory"; DS9: "The Sword of Kahless") The Rozhenko's also had a Human son, Nikolai, who was Worf's foster brother. (TNG: "Heart of Glory", "Homeward") After Worf had his own son, Alexander, and became widowed, he sent his son to live with his foster parents, however, this arrangement was that of a grandson living with his grandparents. (DS9: "Sons and Daughters")
Following the Cardassian withdrawal from the Occupation of Bajor, several of their children were left behind as war orphans. One such case was that of Rugal, who was raised by Bajoran parents, including his foster father Proka Migdal. Unknown at the time, Rugal was a pawn in a spat between Gul Dukat and Legate Pa'Dar. After deliberations made by Commander Benjamin Sisko in 2370, Rugal was taken from his foster parents, who, as Sisko noted, "treated him with love," and allowed him to return to Cardassia. (DS9: "Cardassians")
Endar, a Talarian, took a Human child named Jeremiah Rossa, in accordance with his culture's traditions, after having lost his own son during conflict a with the Federation on Castal I. Following his Talarian custom, Endar took Jeremy, later renamed Jono, as his surrogate son, by claiming the son of a slain enemy. (TNG: "Suddenly Human")
Nicholas Locarno was described by Boothby as being the "coach, surrogate father and best friend, all in one" to the Starfleet Academy cadets of Nova Squadron. (TNG: "The First Duty")
In Klingon, a similar term was soh-chim, the formalized arrangement that assigned one to be responsible for a child in the event of his parent's death. In an alternate timeline, Worf asked Deanna Troi to act as the soh-chim to Alexander, which he described compared to being his surrogate mother. (TNG: "Parallels")