I think Qyburn doesn't directly blame the Lannisters, because Tywin left Harrenhal leaving the Mountain in charge of the place. And Lannisters gave Qyburn several opportunities to climb up the social ladder.
1. Tywin didn't directly tried to kill Qyburn.
When Robb comes around Harrenhal:
ROOSE: We should set the siege lines 1,000 yards from Harrenhal.
ROBB: There won't be a siege. The Mountain can't defend a ruin.
ROOSE: I imagine the Mountain will defend whatever Tywin Lannister tells him to defend.
ROBB: The Lannisters have been running from us since Oxcross. I'd love a fight. The men would love a fight. I don't think we're going to get one.
Tywin disaproved the Mountain's useless brutality (remember when Tywin first came to Harrenhal):
TYWIN: What's this?
GREGOR: We weren't expecting you till tomorrow, Lord Tywin.
TYWIN: Evidently not. Why are these prisoners not in their cells?
GREGOR: Cells are overflowing, my lord.
POLLIVER: This lot won't be here long. Don't need no permanent place. After we interrogate 'em, we usually just...
TYWIN: Are we so well-manned that we can afford to discard able young bodies and skilled laborers? You, do you have a trade?
GENDRY: Smith, my lord.
POLLIVER: What are you looking at? Kneel! Kneel or I'll carve your lungs out, boy.
TYWIN: He'll do no such thing. This one's a girl, you idiot, dressed as a boy. Why?
ARYA: Safer to travel, my lord.
TYWIN: Smart. More than I can say for this lot.
I'm pretty sure with Tywin in charge, working prisoners (Arya, Gendry, Hot-Pie, Qyburn) were treated decently then when Tywin left the place to Gregor Clegane, staffed with working people the Mountain and his men did the massacre on their own.
2. Qyburn had a job at Harrenhal even if he wasn't a real maester.
Here I can't find a real confirmation that he actually worked as a maester, but since Tywin would give a job matching anyone's ability, I suppose he worked as a maester.
3. He had the opportunity to perform surgery on Jaime Lannister.
Qyburn was expelled from the Citadel for performing experiment on living people. With Jaime, he had the opportunity to work on surgery and he did do a great job.
4. After the surgery he went to King's Landing with Jaime and was introduced to the court.
Jaime bring Qyburn with him back to King's Landing and he had the opportunity to meet queen Cersei in person.
5. Cersei Lannister raised him as Grand Maester.
He replaced Pycelle. A new opportunity for his weird experiments. He somehow made Gregor Clegane survive after he was deadly poisoned by Oberyn.
6. He was raised Hand of the King.
Cersei named him as her Hand, the second position of the Realm.