I would recommend building your mother’s tree in the same database. After all, Your Family Tree will be the combination of both branches. One database also means only one place database to build, one set of sources etc. If you need to generate a report of only one branch, this can easily be done using a filter.
But if you still want two databases with only the shared people brought into the new tree…
Create a filter that selects just the people you want. Then do an export. Select Gramps XML (family tree). In the options set the People filter to the filter you created, make sure no living or private information is excluded, and set the Reference option to Do not include records not linked to a selected person. This last options limits places, events, etc to just those items actually needed by the people and families in your filter.
Once you have the export file, you can create the new database for your mother and then import the file.