FXE Tower Controller reported TRACON turned a departure prior to the point designated in Letters of Agreement with Tower resulting in a confliction with traffic in the Tower pattern.

Narrative

A VFR Aircraft X was NW of FXE at 2000 ft. eastbound. FXE is on a west operations and MIA releases Aircraft Y (IFR). I depart Aircraft Y on a 315 heading off Runway 27. On that trajectory Aircraft Y would have passed behind Aircraft X. I call the Aircraft X traffic to Aircraft Y then switch them to MIA. Shortly thereafter; Aircraft X checks in and I immediately call traffic to them; fly them southbound and descend them to 1000 ft. to pry them even further away from Aircraft Y. I then call MIA to let them know what I was doing with Aircraft X.MIA informed me that they were turning Aircraft Y to a 280 heading which put them in conflict with Aircraft X. Bear in mind; according to the LOA between FXE and MIA; MIA has control for turns at 1500 ft. and higher. According to the radar readout; Aircraft Y was at 1300 ft. A traffic alert was issued to Aircraft X and it was turned to avoid Aircraft Y. Aircraft X landed safely without incident. Controllers need to adhere to their respective LOAs with other facilities. Outside of that; communicate/coordinate if they are going to do something outside of the parameters of the LOAs.