Hey! As you guessed, "mother" here is short for the full swear.
A good example is the song "We Had to Tear This Mothaf***a Up" by Ice Cube. You'll typically hear it in terms of "tear this mother up", "tear this mother down", or "tear this mother to the ground". It can be literal, but it also gets used at parties / music events to express that things are going to get wild.
A similar phrase is "we're gonna tear up this town". Meaning we're going to have a wild, reckless amount of fun.