You file one tax return for the entire year. If the LLC has more than one member, you file a form 1065 partnership return for the LLC, which issues a K-1 to each member, and the member puts the K-1 on their personal tax return and pays the taxes. (The only time an LLC does not have to file a form 1065 is if the only two members are spouses and you live in a community property state.)
So the taxes on the LLC income are really due on each member, not the LLC itself. And the members do have to pay estimated taxes. You don't "file" a quarterly return, but you are required to make quarterly payments. Form 1040-ES can be used to calculate the estimated payment, but you don't send the 1040-ES to the IRS, you keep it for your own records, and only send your payment.