If you have a written lease with a “Waiver of a Notice to Quit” clause, it only applies to "Lapse of time evictions" and not other types of evictions such as non-payment...see the statute below.

The Connecticut General Statute Sec. 47a-25, Waiver of notice to quit says;

When, in any written lease of any land, building, apartment or dwelling unit, notice to quit possession has been expressly waived by the lessee in the event such lease terminates by lapse of time, the three days' notice prescribed in sections 47a-23 and 47a-23a shall not be necessary; and complaint and summons may issue in the same manner as if such notice to quit had been previously given.