Broken speaker connector on motherboard

No, not necessarily.

If you can determine the make of the connector or find a compatible alternative, you may be able to solder on a replacement. You should check to see if the traces are damaged or have been stripped from the board. If that isn't the case, replacement as a procedure should not be very difficult. If your connector is also in good condition without damaged contacts, you might even be able to re-solder it onto the board without needing a replacement at all.

For the matter of sourcing a replacement connector, it's a bit difficult for me to find an alternative myself. Despite this, I can see from the motherboard that the cooling fan connector may be the same as the speaker connector. If you can verify this, it gives you another avenue for searches when looking for parts. Unless there's a part number, things might need to get a bit more technical for isolation of an alternative.

The observable characteristics that may apply are that it seems to be the following:

• FFC/FPC (Flat Flexible)
• 4 point/4 pin
• Surface Mounted
• Vertical
• Pins/Contact at 1 side
  

Measurements and more particular detail might be necessary to really drive it home when purchasing online through specialized component retailers, but that's the start. If you can find a local shop that sells motherboard replacement parts or the like, you might be able to get it through them or at least have someone experienced isolate the part for you.

Hope this helps.