Im not sure if you want to remove or replace the apostrophes in the select. There is a replace function in SQL replace(string, replacestring, replacewith)
SELECT replace('test','est', 'aste')
returns
Taste
So depending on the fields you are using you can strip out the apostrophes on the select of the trigger.
SELECT
@PassengerName=replace(PassengerName,'''',''),
@DocumentType=replace(DocumentType,'''',''),
@BoardingSequenceNumber=replace(BoardingSequenceNumber,'''',''),
@FlightNumber=replace(FlightNumber,'''',''),
@CarrierCode=replace(CarrierCode,'''',''),
@DepartureTime=replace(DepartureTime,'''',''),
@Destination=replace(Destination,'''',''),
@DeviceAddress=replace(DeviceAddress,'''',''),
@Timestamp="Timestamp",
@WorkstationId=WorkstationId,
@DeliveredTimeStamp=DeliveredTimeStamp
from inserted;
If you don't want to remote the apostrophes you could double them up in the same way.