There is no direct connection from St. Julian's to Marsaxlokk. However, you can take the bus to Malta Airport then take the drive to Marsaxlokk. Alternatively, you can drive from St. Julian's to Marsaxlokk in around 14 min.
How to get from St. Julian's to Marsaxlokk
From St. Julian's bus to Malta Airport then drive to Marsaxlokk. 33 min 8€. From St. Julian's drive to Marsaxlokk. 14 min 1€.