investment in IRA , in what month is the deadline for conversion?

There is no deadline.    You can convert a Traditional IRA to a Roth anytime you want.    It will be reportable in the tax year in which the conversion occurred.    If converted in the 2020 calendar year it will be reported next year when you file your 2020 tax return.

The deadline for contributions to regular IRAs is July 15, 2020 for tax year 2019.  For certain self-employed taxpayers who file an extension, contributions to self-employed retirement plans (such as SEPs and Individual 401(k) plans) can be made until October 15, 2020.   There is no deadline for conversions of existing IRAs.