If I quit the company, what will happen to the municipal tax and metropolitan inhabitant tax that have been deducted from my salary?

If the person in charge at the company or office where you worked submits a transfer notification to the Municipal Tax Section of the Taxation Division, the payment will be switched from salary deduction to individual payment. After that, the Citizen's Tax Section of the Taxation Division will send a tax notice (payment notice) to your home, so please pay by the deadline.
In addition, if you collect municipal tax and metropolitan inhabitant tax for the year at the time of retirement, you will not have to pay municipal tax and metropolitan inhabitant tax for that year.