No, using Passenger cards to move a Passenger between two cities on an opponents route does not allow you to add points for completed Destination Tickets at the end of the game.
In the rule book under Calculating Scores (page 6) and Drawing Destination Ticket Cards:
Players should then reveal all their Destination Tickets and add (or subtract) the value of their Destination Tickets still in hand, based on whether they successfully connected those cities together or not.
Each Destination Ticket includes the name of two cities on the map and a Point Value. If a player successfully completes a series of routes that connect the two cities, he will add the amount of points indicated on the Destination Ticket to his point total at the end of the game. If he does not successfully connect the two cities, he deducts the amount of points indicated.
You connect cities by claiming routes (Page 5)
A route is a set of continuous colored spaces (in some instances, gray spaces) between two adjacent cities on the map. To claim a route, a player must play a set of Train Cards whose color and quantity match the color and number of spaces of the chosen route.
Passenger cards (10 total) are not used to claim routes, they are used during a Game Turn which a player chooses to 4. Move Passengers (page 3). The Passenger card allows you to move a Passenger through an opponents route.
Passenger Cards can be drawn like any other Train cards. When moving a Passenger, the player can play any number of Passenger Cards and use one route belonging to an opponent for each Passenger Card played.