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Have a look on VideoCapture::get, where using the property identifier CV_CAP_PROP_FPS you will get the frame rate, using this you can,

• Convert start time and stop time to frame count using

   start_frame_count = frame_per_sec * 60 * start_time_in_min;   
   stop_frame_count = frame_per_sec * 60 * stop_time_in_min;
  

• Count frames in while loop.

• Check whether the count reaches on start or stop frame count.

  The code look like,

   frameCount=0;
    while(1)
    {
     frameCount++;
     capture.read(frame); 
     if(frameCount>=start_frame_count&&frameCount<stop_frame_count) 
        VideoWriter.write(frame); 
     else if(frameCount == stop_frame_count)
       break;
     }
  

Also refer the answer here, might be useful.

Note: The code is not tested.