Is it permissible for a traveler to shorten the prayer (qasr) if the time for the prayer began before he departed from his town, and he commenced his journey after the time began and before the time for the next prayer entered?

If you did not pray the prayer on time while you were a resident (muqeem), then prayed it while you were a traveler (musafir), pray it shortened (qasr). This prayer is considered adaa' (performed on time), not qadaa' (performed as a make-up), because its time is still remaining. Prayers performed during travel are prayed according to the specific manner prescribed for prayer during travel, just as prayers performed while resident are prayed according to their specific manner.