According to the SBB website you would get a Swiss Family Card for free, and your daughter would travel for free if the accompanying adult has a valid Swiss Travel System ticket (Swiss Pass, Swiss Holiday Card, etc.). You can get this card either at the ticket office, but you also should be able to get it on-line (if you book your Swiss Travel System ticket(s) online, you have the option to add the Swiss Family Card), so it depend on what you are getting or already have.
A note: the SFF website does not know about a "Junior Travelcard"; it may have been something from the past.
And a second note: you may have to be careful with the ticket/pass/card you get, because particularly on mountain railways, there may or may not be a discount. You might ask again when you have an itinerary (unless that is against "the rules"…).
As it has been said in another comment, there may be lines, but they should move relatively quickly; you might aim for a counter stating "Abonnemente" or "Freizeitangebote" (keep in mind, it is two years that I was last at Zürich Airport station, so, things may have changed); otherwise, just line up. If it is just for the Swiss Family Card, it should not take thaaat long.
Also, don't worry about not catching a particular train; depending on where you are going, you will have another connection half an hour or at least an hour later.
Otherwise: Enjoy your time in Switzerland!