(I'm a US native speaker). Both are fine. "Quieter" is more common. They mean the same thing.
As far as I know, "more X" is fine, even if "X-er" exists.
A quick Google search of Project Gutenberg turns up many examples of both forms:
"I really do eat better, and am more quiet than I was."--Charlotte Perkins Gilman
"Sahwah was more quiet, and there was a sober look in her eyes."
"He had fallen back in his chair, not looking at her now, and with his hands, from his supported elbows, clasped to keep himself more quiet."--Henry James
"a barbarous aberration, from which other more quiet nations abstained..."--Otto Jespersen
"Jane was as much gratified by this as her mother could be, though in a quieter way."--Jane Austen
"The horse was bought; nothing could go quieter; Evelyn was not at all afraid."--Edward Bulwer Lytton
"Then her own life took a quieter turn; we met again; I went for a good while often to her house."--Henry James, who also used "more quiet," above.
"I think the baby will be quieter if I put him the other way."--Wilkie Collins