No, you can claim an education credit regardless whether or not you started paying your loan. To qualify, you must have a 1098-T that reports an amount on Box 1 of the form. Right now, the loan is what is making the payments. You will need to pay it back eventually so you might as well receive credit now.
There are other conditions that apply to claim the credit and Turbo Tax does a very good job in asking those questions to determine your eligibility for the credit.
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