The canon, by direct reading and by inference, does not seem to suggest so:
* Even though he knew of <PERSON>'s relationship with <PERSON> and the fact that she was pregnant, there is nothing to suggest that <PERSON> was aware that <PERSON> was pregnant with twins, as opposed to one child, in the first place
* I couldn't find any references to <PERSON> either knowing, or using that knowledge to some purpose
* The absence of that knowledge appears to have specifically hurt <PERSON> in ways that he could have reasonably avoided and should have wanted to avoid
* <PERSON> never pursued an investigation into it (and did not seem to know anything about it at all until the end), though others did, and there is no evidence that <PERSON> learned about it through those other investigations
Twins
There is no evidence that I can find in the canon that anyone other than <PERSON><PERSON>, <PERSON> and <PERSON> (and possibly <PERSON>) knew that <PERSON> had twins. Her pregnancy was not a secret but she did not reveal to <PERSON> that she was carrying two children, if indeed she herself knew, only that she was pregnant. Therefore, unless the Emperor knew specifically about both of them then he and anyone else who was looking for them would probably only be looking for one child.
Direct answer
I could not find any evidence in the canon that <PERSON> knew that <PERSON> had successfully given birth before dying, either when it happened, or afterwards. The answers in this question and this question suggest that he did not know at the time or at any time after that, based on how he behaved after <PERSON> came out into the open. See also here and here for reference.
Answer by inference
There is also no evidence that <PERSON> actually did anything about it, if indeed he even did know; there were no programs of spying or clever stratagems. In fact he suffered repeatedly precisely because he did not apparently know about this. 950
<PERSON> was a public figure who did not hide (most of the time) and <PERSON> never left Tatooine after arriving there (until ANH), so keeping tabs would not have been difficult, and yet:
* If <PERSON> had been aware of the birth he would have been aware of <PERSON>'s location on Polis Massa. <PERSON>, <PERSON><PERSON><PERSON> and <PERSON> were all present there and yet <PERSON> shows making no effort to pursue them or investigate Polis Massa itself.
* If he had known about <PERSON>, his surveillance on <PERSON> would have been far greater including his role in the Rebel Alliance. Although <PERSON> was aware of the budding Rebel Alliance through <PERSON>, and knew of <PERSON>'s anti-Imperialism (going back to the Clone Wars days) plus his involvement in the Corellian Treaty incident, he did not appear to be aware that <PERSON> was actively involved until too late. <PERSON> only participated in the Corellian Treaty via holotransmission and was not seen by <PERSON>.
* Although <PERSON> was known by that time to be somewhat anti-Imperial (she had been detained earlier on Kashyyyk), her role in the Alliance was not known even though her father's role was. In addition, <PERSON> was physically present at a later meeting on Kashyyyk to formally establish the Alliance wherein the Emperor could have found <PERSON>, <PERSON>, <PERSON>, <PERSON> and <PERSON>, all key members in the Rebellion that the Emperor had helped create and knew was going to be a big problem (TFU ending).
* If he had known about <PERSON>, he would have been able to prevent the theft of the Death Star plans or at least done far better in attempting to prevent it. Even though the Death Star was known to the Rebel Alliance at that time, he did not know that <PERSON> was necessarily a part of it and too late did <PERSON>/<PERSON> attempt to stop her as discussed in my answer here.
* If he had known about <PERSON>, he would have had an opportunity to intercept the stolen Death Star plans. <PERSON> presumably knew at some point that <PERSON> had been intercepted over Tatooine, and <PERSON> never left Tatooine after arriving there, so if <PERSON> put a) the stolen plans, b) <PERSON> and/or <PERSON>, and c) Tatooine together, the Empire's attempts to retrieve the plans should/would have been far more effective. 865