No, if you are having extra taxes withheld from your regular W-2 job to cover your self-employment tax obligation, you would not need to make estimated payments too.
To answer your other question though, the first quarter estimated payment due date was changed to July 15, 2020. The second quarter estimated payment due date has also been extended. Instead of being due by June 15, 2020, it is now due by July 15, 2020.
[edited 4/13/2020 | 8:10 am PST]
@JR_TT
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