Like Dave I only have the manual wind version of the SQ-A, but obvious possible culprits are the batteries. There is a manual here: (http://www.butkus.org/chinon/bronica...onica_sq-a.htm) which is probably the one you found already but I list it just in case... There is says:
G. Upon loading the film, set the mode selector switch to Sm or Cm and depress the drive button, which will place the first frame into position for taking the picture.
It also says you can wind the film to the 1st frame manually (obviously it is better to use the motor):
H. When the film back is detached from the camera main body for film loading, the manual film winder is used for advancing the film. The film will stop when it is in place for the first exposure, with the exposure counter also changing to "1 ". However, it will be possible to rotate the manual film winder and, therefore, it should be rotated 2 or 3 times more, in order to take up any slack in the loaded film.
Other possibilities are to play with the camera settings: try it with the darkslide inserted and again when removed. Try with and without a lens attached. Also check that you don't have the mirror-up lever engaged; ditto the multi-exposure lever: the SQ-A can be temperamental about such things...

Sorry if you have already tried all this....

Good luck, Bob.