It's correct but odd. "The sons do not rely on the girls" would better be translated as "Sønnerne lægger ikke deres lid til pigerne".
It depends on the construction you use in a particular language. In dutch we can use something similar, but it leads to a subtle difference in meaning. The literal translation here is "(ze) vertrouwen niet op (på) de meisjes", and I would translate that as "they do not trust/depend on the girls". If you use "(ze) vertrouwen de meisjes niet" (not using 'på' here), it would translate to "they don't trust the girls".
Tl;dr in Dutch there is a subtle difference betweeTurksen 'vertrouwen' (trust/depend) and 'vertrouwen op' (Trust in). Maybe it's the same in Danish, maybe not.
In modern English you need the auxiliary verb "do" to form negations (except when the main verb is "be" or another auxiliary). Therefore the correct translation of "Sønnerne stoler ikke på pigerne" is "The sons do not trust the girls", although it is kind of "tempting" to use the Danish syntax when you are translating the sentence from left to right and suddenly you encounter "ikke" and have to rearrange everything :p