This error is due to the quota for the NCv3-series having a default of zero:
For this size family, the vCPU (core) quota in your subscription is initially
set to 0 in each region. Request a vCPU quota increase for this family in an
available region.
If your quota is zero because you are on a free subscription such as an academic sponsorship, Microsoft will probably keep the quota at zero to save these high-demand machines for paying customers.
One quick solution is to use NV6 instead of NC6_v3. The NV-series uses Tesla M60, which Wikipedia lists as several generations old compared to the latest V100, but has no quota limitations. After choosing the cloud image on Azure, click "Change size", and change to NV6.
If you want to request an increase in quota, first confirm that the NCv3-series is available for your region, e.g. for Europe and UK go here.
Then, request a quota increase following the official Microsoft directions, or the directions below, which are quicker and more up-to-date:
Follow the link in the error message (starting with https://aka.ms), or go to portal.azure.com, click on "Help + support" in the left blade, then
on "New support request" in the blade second from the left
Answer the details in "1. Basics":
Issue Type: "Service and subscription limits (quotas)"
Subscription: if you can choose, select the subscription ID in the
error above.
Quota type: "Compute/VM (cores/vCPUs) subscription limit
increases"
support plan: you should have "Azure Support Plan - Developer"
selected
click "Next"
Answer the details in "2. Problem":
Severity: keep the default "C - Minimal impact"
Quota details:
Deployment model: keep the default "Resource manager"
Location: choose a region that has NCv3-series
SKU family (see meaning here): choose NCSv3. In the documentation, Standard NC6s_v3 is listed under "NCv3" series, which is not listed as an SKU family. The NC family has a default quota of 48 and the NCSv3 has a quota of zero, so I assume that NCSv3 is the right SKU family. (I raised a ticket with Microsoft asking to either change the form drop-down or the documentation.)
New limit: "6"
click "Save and continue"
Click on "Next", enter your contact details, then create the request.
Then you need to wait for Microsoft to increase your quota.