Read As Written, yes
While the implication of the description is that the character has direct knowledge of the fact or secret, that does not necessarily preclude the nothic from learning something that might be buried in the character's subconscious, nor is it explicitly stated that direct knowledge is what is able to be gained.
If the nothic wins, it magically learns one fact or secret about the target.
A "fact" is just something that is true, and since it does not explicitly state that it learns something the character has knowledge of, this opens up the door for interpretation as to how the Nothic is able to come by this knowledge.
Perhaps the Nothic could search through memories and "see" the faces of the people that abandoned the character, even though the character cannot remember it themselves.
Perhaps the Nothic is looking into the character's past, not into the character's mind, and is able to view events as they unfolded.
Further indication that it may be able to extract information not known to the target is in the the description of Nothics in the MM (p. 236) (emphasis mine)
Dark Oracles. Nothics possess a strange magical insight that allows them to extract knowledge from other creatures. This grants them unique understanding of secret and forbidden lore, which they share for a price. A nothic covets magic items, greedily accepting such gifts from creatures that seek out its knowledge.
The specific reading of this is:
Nothics... extract knowledge from other creatures -MM pg. 236
Which is not the same as:
The Nothic is able to extract the knowledge of that creature
The Nothic is able to extract another creature's knowledge
If either of the two above examples were used, then it would explicitly mean that it could only extract something known to the target. As the actual text reads, it never specifies this. The actual text is very general and does not indicate that the knowledge learned needs to be possessed by the target, whereas the two examples are far more specific and denote knowledge the target actually holds.
Read As Intended, possibly not
Since there is no further clarification other than the description of the ability and the creature itself in the MM, one could assume that this is intended to work like detect thoughts in that the target must be actively conscious of the information being gained, though this is not supported anywhere.
Ultimately I would suggest discussing with your players about whether they would be agreeable to such an outcome, some might think it would be a neat plot hook while others would prefer the mystery of not knowing.
While not explicitly relating to this situation, in a game I am currently running I have a player who enjoys when I add things to her backstory that she as a player and as a character was unaware of through dialogue with NPCs (due to having her memory wiped) and then having an opportunity to act on the new information, but this does take a level of trust between you as the DM and them as the player.
Another player at the same table prefers to discuss in detail things relating to their backstory that they would like incorporated into the story before actually moving forward with it.
When it comes to something like this, to save myself time and headache I will often default to the lowest common denominator to make a table ruling; since one player would not be amenable to new information being sprung on them, I would simply not have the Nothic able to gain information outside of character knowledge even though it is not explicitly disallowed.