So, we know that Secret EGO has had run-ins with ID's elderly stowaway. Has there ever been a case where the League has crossed paths with Turquoise? Are they aware of her existence at all? Knowing that EGO has precious cargo would be a very unfortunate piece of leverage, if they're willing to stoop that low.
As of right now, no, the League is not aware of Turquoise's existence, nor have they really encountered her before! And if they ever have, they wouldn't give her the time of day because she isn't the focus of...whatever they're up to. The League of ID does not have a lot of information on Secret EGO as an organization, much less the whole list of people who work there/happen to be stationed there for safety purposes - it's a huge secret agency, after all. And EGO would presumably do everything in its power to keep Turquoise safe.
...On that note, however. If the League does cross paths with Turquoise, with the additional information that this particular pipsqueak is valuable to (at least) one of their most notable opponents...I wouldn't be surprised if they end up taking the opportunity.