Although large language models have transformed natural language processing, their computational costs create accessibility barriers for low-resource languages such as Brazilian Portuguese. This work presents a systematic evaluation of Parameter-Efficient Fine-Tuning (PEFT) and quantization techniques applied to BERTimbau for Question Answering on SQuAD-BR, the Brazilian Portuguese translation of SQuAD v1. We evaluate 40 configurations combining four PEFT methods (LoRA, DoRA, QLoRA, QDoRA) across two model sizes (Base: 110M, Large: 335M parameters). Our findings reveal three critical insights: (1) LoRA achieves 95.8\% of baseline performance on BERTimbau-Large while reducing training time by 73.5\% (F1=81.32 vs 84.86); (2) higher learning rates (2e-4) substantially improve PEFT performance, with F1 gains of up to +19.71 points over standard rates; and (3) larger models show twice the quantization resilience (loss of 4.83 vs 9.56 F1 points). These results demonstrate that encoder-based models can be efficiently fine-tuned for extractive Brazilian Portuguese QA with substantially lower computational cost than large generative LLMs, promoting more sustainable approaches aligned with \textit{Green AI} principles. An exploratory evaluation of Tucano and Sabiá on the same extractive QA benchmark shows that while generative models can reach competitive F1 scores with LoRA fine-tuning, they require up to 4.2$\times$ more GPU memory and 3$\times$ more training time than BERTimbau-Base, reinforcing the efficiency advantage of smaller encoder-based architectures for this task.