OK, let's put it another way.
As E-series documentation declares, there are two triggering actions associated with each counter: Start trigger and Pause trigger.
How to connect these two triggers with two PFI lines (if it is possible with E-series DAQ card)?
Please take a look at this example from the Example Finder: Meas Two Edge Separation. You can use the Gate and Source of a single counter to measure the time between a rising (or falling) edge of a signal with respect to the falling (or rising) edge of another one. This example can be found in Help » Find Examples » Hardware Input and Output » DAQmx » Counter measurements » Two-Signal Edge-Separation.