The short answer is no.
In the beginning, it was common if not recommended practice to create a locator as a database object, but a few years back that functionality was dropped. I think the reasoning was based on the idea of publishing geocoding services, although that's just a guess on my part.
Why do you want to store the locator in SQL Server database? If you have Enterprise, you can publish the locator to Enterprise portal then access the the locator item in Pro from My Content on the portal. Or you can set the published locator item as a geocoding utility service that will be available in ArcGIS Pro whenever you sign in and your portal is the active portal.