Hola
Puedes usar un QSignalMapper:
http://doc.qt.io/qt-5/qsignalmapper.html
Algo así:
Código C++:
Ver originalsignalMapper = new QSignalMapper(this);
// conectar los botones al mapa
connect(ui->pushButton_1, SIGNAL(clicked()), signalMapper, SLOT(map()));
signalMapper->setMapping(ui->pushButton_1, 1);
connect(ui->pushButton_2, SIGNAL(clicked()), signalMapper, SLOT(map()));
signalMapper->setMapping(ui->pushButton_2, 2);
connect(ui->pushButton_3, SIGNAL(clicked()), signalMapper, SLOT(map()));
signalMapper->setMapping(ui->pushButton_3, 3);
connect(ui->pushButton_4, SIGNAL(clicked()), signalMapper, SLOT(map()));
signalMapper->setMapping(ui->pushButton_4, 4);
connect(ui->pushButton_5, SIGNAL(clicked()), signalMapper, SLOT(map()));
signalMapper->setMapping(ui->pushButton_5, 5);
connect(ui->pushButton_6, SIGNAL(clicked()), signalMapper, SLOT(map()));
signalMapper->setMapping(ui->pushButton_6, 6);
// conectar el mapa con el SLOT
connect(signalMapper, SIGNAL(mapped(int)), this, SLOT(botonPulsado(int)));
Luego en el SLOT recibes el código que asignaste a cada botón, por lo que sabes cual se pulsó.
Saludos