This is what is called an amortization problem. It is usually solved with a table or an iterating program. Here is a way to get the approximate answer directly. The principal due at the beginning is 29885-3420 = $26465. Over one year, with a declining balance, the average balance is $13232 and the interest on that amount over one year will be $2117.
It the loan is paid off in equal quarterly amounts, they must add up to 26465 + 2117 = $28582. The quarterly paymewnts should be 1/4 of that, or $7146.
Balance due at start (after down payment):
After one quarter: add 1058.60 for interest due and subtract 7146 principal payment. Remaining balance = 20,377.60
After first quarter: add $815.10 interest and subract 7146 principl. Remaining balance = 14,046.70
After third quarter: add 561.87 interest and subtract 7146. Remaining balance = $7462.57
After fourth quarter: add $298.50 interest and subtract 7146. Remaining balance = $615.
For a second iteration, I would recommend adding $154 to each quarterly ayment, to get rid of the $615 deficit on the first attempt.
That makes the quarterly payment $7300 after one iteration. I end up overpaying $38 this way, so the third iterated answer is $10 less per quarter, or $7290.
There is a handy amortization calculator at this web site:
Enter the initial loan balance of 26465, the 16$ interest rate, and 4 quarterly payments.
Using it, with quarterly compounding you should get an exact loan payment of $7290.84