You can't apply for the CSQ until you've gone through the stage one CIC process. That took about 8 months for us. The Quebec process is pretty fast though (51 days for us) and they are WAY more helpful than CIC when it comes to answering questions.
Hi how much time does it take after i submitted my expression of interest for the Quebec Skilled Worker Program? Adding to this could you let me know can we add the language skills score after submitting the EOI?