I am trying to run following SQL:
INSERT INTO Suppliers ( [SupplierID], [CompanyName])
Select [SupplierID], [CompanyName] From [AlexDB]..Suppliers
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You can easily add a "Linked Server" from SQL Management Studio and then query on the fully qualified table name. No need for flat files or export tables. This method also works for on-prem to azure database and vice versa.
e.g.
select top 1 ColA, ColB from [AZURE01_]._UAT_RecoveryTestSrc.dbo.FooTable order by 1 desc
select top 1 ColA, ColB from [AZURE01_]._UAT_RecoveryTestTgt.dbo.FooTable order by 1 desc