It is possible! The resulting fruit will be a hybrid.

The seascape strawberry (Fragaria x ananassa "Seascape") is a hybrid ever-bearing strawberry variety. Strawberry plants produce runners that put down roots and form new plants. You can propagate strawberries by planting the runners, from seeds or by the roots.

You can pot the mother strawberry plant and wait for runners or stolons and put small mounds of dirt around the 'daughters'. That will give the Seascape strawberry.

However, since this is a hybrid, the seeds may not grow true to the Seascape taste or size. You can grow the seeds into strawberry plants. Wait until the strawberry is starting to become very mushy and dry the seeds from it.

Here is an article regarding starting strawberry seeds: Easy Tips to Grow Strawberries from Seed.