Berlin to Hamburg with Quer-Durchs-Land-Ticket?

Sure, why not? Just use the regional trains.

When looking for your route and time schedule on the Deutsche Bahn website, click on the box that says local trains, and unclick the box that say fast trains like the ICE or IC trains. This trip will take you between 3.5-4 hours using the regional trains.

You might want to rethink your plans.

A Quer-durchs-Land-Ticket will cost 76€ for the five of you (no child-under-15-free on QdLT) and it will take 4 hours unless you go part way by bus (bus fare not covered by QdLT).

On the other hand, if you can commit to a specific connection (Time & Date), you might be able to find a Saving Fare as low as 89€ for all of you (4 adults and one under-15 child). For June 11, I see a direct EC connection for 89€ taking 2 hours.

If you book now, you can get the bus for 9 euro per advance non-refundable time specific ticket, trip is a bit over 3 hours.

The only good thing about regional trains is the stopover in Schwerin, which gives you the chance to see the Neuschwanstein of the North and the beautiful old town.

By the way, if you can do this trip on a weekend day, it's only 44€ with a Schönes-Wochenende-Ticket.

Thanks all - super useful!!

The bus option looks good at 9 EURO person - checking the bus station locations whether these are convenient spots.

Presume I can book through email for the e-tickets?

This is the site right?

https://www.berlinlinienbus.de/bus-berlin-hamburg

Yes, that's the site. Tickets seem to be 90 days out, so on your travel date you should be able to do this starting Monday.
I don't see any reason why you can't do this over the internet with a credit card, although you may need to call the credit card company for the purchase to go through.
You should be able to locate the bus stations easily on the Berlin map. Note that depending on the time you want to go the bus will be going from either the Funkturm or the Ostbahnhof stations, but not through both (that I can tell). Both are easily reached via the transit lines.
If this gives you peace of mind, we are doing this also, but first week in July, I can't purchase our tickets until next month. So I can't specifically comment on the ease of use of the site, but I can't believe you will have any problems.
Enjoy.

Whether you go by bus or train, you might want to look at what you are getting. Both take almost the same amount of time. Train is a bit more expensive, but not by a lot. Have you seen the size of the bathrooms on a travel bus? Train toilets win out over bus toilets every time. You are allowed to get up and stroll around the train, you can't do that on a bus. You are pretty much stuck in your seat for over 3 hours. So, for me, the train is the winner. The difference in price doesn't warrant choosing the bus.

And for us the price is most definitely worth the difference. The bus runs more often, and in so many places faster, more direct, and more convenient that the train. We think nothing of a 3+ hour car ride to get somewhere, and that entails one of us actually driving. Is sitting in a car for 3+ hours different than sitting in a bus for that time, especially when the car does not have a bathroom?