Hi Abraham,
You need to create a virtual network link from the private dns zone to the peered virtual network so that when the VM does DNS lookup for your Azure SQL Server it will receive the IP address of the private endpoint instead of the public address.
To create virtual network link, browse to the Private DNS Zone that was created for your private endpoint -- Virtual links blade. Click Add, enter a name for the link, select the peered virtual network, click Create.
It may take a minute or so for it to finish creating the link, so wait for the success notification. Once it has been created, using command prompt, do nslookup of the FQDN (yourservername.database.windows.net) from the VM and make sure it is returning the private IP address instead of public.
nslookup yourservername.database.windows.net
Resolution virtual network
Please click Accept Answer and upvote if the above was helpful.
Thanks.
-TP