The network scenario offers interesting new perspectives on the phenomenon of quantum nonlocality. Notably, when considering networks with independent sources, it is possible to demonstrate quantum nonlocality without the need for measurements inputs, i.e. with all parties performing a fixed quantum measurement. Here we aim to find minimal examples of this effect. Focusing on the minimal case of the triangle network, we present examples involving output cardinalities of 3-3-3 and 3-3-2. Finally, we discuss the prospects of finding an example of quantum nonlocality in the triangle network with binary outputs, and point out a connection to the Lovasz local lemma.