Is it permissible to fast on Mondays and Thursdays with the intention of quitting smoking, without going against the approach of the Prophet, may Allah’s prayers and peace be upon him?

If "fasting all days" means continuous fasting without breaking it except on prohibited days, then the preponderant opinion is that it is disliked (makrooh). However, continuous fasting for part of the year is permissible, as it is affirmed by the Prophetic Sunnah. The Prophet (peace be upon him) sometimes fasted continuously and sometimes broke his fast. The best fasting is the fasting of Dawud (peace be upon him), who used to fast one day and break his fast one day; this is the most balanced form of fasting. A Muslim must choose what they are capable of from voluntary acts of worship and persist in them without undue hardship. Quitting smoking is a good thing, and fasting helps with it.