Can I substitute quark (40%) for cream cheese in basque burnt cheesecake recipe?

When using quark instead of cream cheese, the main issue is that quark is overall “wetter” - especially when the recipe uses US cream cheese, which is pretty much a “brick” compared to the (same brand) product in Germany. You can mitigate that by placing the quark in a cloth-lined sieve overnight to drain it, but that’s optional. I find that most recipes that use quark instead of cream cheese are a tad crumblier, cream cheese fillings are a bit smoother (if you use a good brand).

That said, there’s a huge tradition of quark-based cheesecake in Germany and other European countries, and as long as your recipe contains enough binding agents, i.e. eggs and/or flour or starch, you will be fine. Looking at your recipe, it uses six eggs. That’s on the generous side even for a regular quark-cheesecake. So I’d say just go for it. If you are super worried, you could add a tablespoon of cornstarch, just in case, but I don’t think it’s necessary. I would suggest that you do a quick Google search for “Käsekuchen ohne Boden“ (crustless cheesecake) as that’s the German cousin of the basque cheesecake. You will see that the recipes are quite similar, just with quark and maybe a bit of flour, starch or semolina.