Unfortunately the Vertex Search filtering functionality is pretty limited at the moment.
Here is the documentation for how to implement filtering using the API. You'll notice there is no built-in support for date filtering currently, but there is support for filtering on numerical fields.
Workaround:
One workaround I can think of right now is to ingest documents with their dates converted to Unix timestamps (integers). In the front-end of the application you could provide a normal date picker to provide a date range, and then in the backend convert this to unix time and perform numerical filtering.
Example: "Jan 12 2023 to Jan 12 2024" would be equivalent to a unix time filter of between 1673481600 and 1705017600.
In Vertex Search this would be applied as a filter in the syntax: date_field_name: IN (1673481600, 1705017600)