It depends on the specifics of the train ticket and how far you need to go on the bus.
There are no trains going to Billund (airport), so you'll have to take a bus from somewhere, pouplar choices include Vejle and Give, but in any case I believe DSB is the only train company operating on the route. Their regular (not "Orange") tickets are normally valid for connecting buses in some "fair" area around the train station. I don't know the details for how the buses work in Odense, so I can't say if there's an exception there, or whether Odense SQ (wherever that is, it means nothing to me) is within the "fair" area, but if it's just 1 zone, my guess would be it is valid. At the other end of your journey, my guess would be that Billund airport is too far from either of the possible train stations for your ticket to be valid for the whole bus journey.
I guess this might be a situation where the rejsekort actually makes things easier, but I'm still not a fan so I don't know enough about it.