From the information I’ve found from June 14, 1928 to July 1st they made about 3500 model Q cars which they called the 1928 model Q and then from July 1st to January 2nd 1929 they made about 62,000 more cars which they called the 1929 model.
Sure seems to me that this car would have been made in that first week of production and the end of June 1928 with this engine being #1265
(Q176265)