What rumors were Cersei and Qyburn talking about in “No One”?
We don't know yet.
Is there any clue about the rumors in previous episodes?
No, or maybe yes if we count Bran's vision a hint.
Bran does see a vision of green fire aka wildfire, which is one of the most talked about rumors
[source: bustle.com, reddit].
Even the dialogue looks suitable to the rumor:
Cersei: And? Was it just a rumor or something more?
Qyburn: More. Much more.
From the same bustle.com link, a few more suggestions for the rumor are:
- Since Varys' little birds are known to travel all over the place, the rumor could have something to do with Essos. If that's the case, it could be about Daenerys' dragons — or the fact that her younger brother is working with Daenerys. Either way, intel on Daenerys is pretty powerful in itself, and maybe she'd contact the Mother of Dragons to form some sort of alliance.
- But what if Cersei learned her brother had nothing to do with it, and was investigating the Tyrells for Olenna's part in the murder? If Cersei found out that the Tyrells killed Joffrey, after Margaery convinced Tommen to accept the High Sparrow and Lady Olenna abandoned her at King's Landing, her vengeance would be unparalleled. If she told Tommen that Margery had something to do with Joffrey's death, there's a chance he could turn against his wife — and the Faith of the Seven — which could lead to him securing the kingdom for his family once again.
But all three are just rumors for now, and the wildfire rumor seems more reasonable as her reaction was not filled with anger. As per her nature, if she got to know about Margery's involvement in his son's death she might have lost her mind and not waited this long to act. On the other hand she hates Tyrion and doesn't look like a person who would make an alliance with the Targaryens, as they are enemies of the Lannisters.
In GoT S06E10, we got to know that rumor was about: