"The certificate chain was issued by an authority that is not trusted" when connecting DB in VM Role from Azure website

You likely don't have a CA signed certificate installed in your SQL VM's trusted root store.

If you have Encrypt=True in the connection string, either set that to off (not recommended), or add the following in the connection string:

TrustServerCertificate=True

SQL Server will create a self-signed certificate if you don't install one for it to use, but it won't be trusted by the caller since it's not CA-signed, unless you tell the connection string to trust any server cert by default.

Long term, I'd recommend leveraging Let's Encrypt to get a CA signed certificate from a known trusted CA for free, and install it on the VM. Don't forget to set it up to automatically refresh. You can read more on this topic in SQL Server books online under the topic of "Encryption Hierarchy", and "Using Encryption Without Validation".

If you're using SQL Management Studio, please goto connection properties and click on "Trust server certificated"

If you're seeing this error message when attempting to connect using SSMS, add TrustServerCertificate=True to the Additional Connection Parameters.