I have not personally used the DAQCard 1200, but if it is similar to the E-series devices, you route the trigger signal to the gates of both counters.
In your code, use the Data Acquisition->Counter->Generate Delayed Pulse VI twice, once for each counter. You can then set an input on that VI to select to trigger on the gate, and set the delay and pulse width of the pulse. Since both will start from the same trigger, you can delay either pulse as long as you wish, from microsecond to seconds, making the second pulse follow the first.
Mark