For coupons, which are usually for a stated percentage of the face value of the
instrument, the daycount convention does not matter unless the coupon is paid late
or the bond is a callable bond that is called in between two coupon payments. That
is, it is only when interest needs to be computed for a period that is less than
the interval between two coupon payments that the daycount convention matters.
Edit in response to additional comment by OP: The running yield changes from day to day depending on the daycount convention, but if the coupon is for 1% of the face value payable on the last day of each quarter, then 1% is what you get on the last day of each quarter. What you perceive you got as the daily rate over the quarter just ended depends on which quarter you are talking about since they have different numbers of days. Unless the face value is in the order of hundreds of thousands, the numerical difference is not worth bothering about.