Under the banner of NATO, not that I'm aware of.
However the forces of NATO nations under UN command have fought directly against Russian forces in the Korean War - specifically pilots flying MiG-15's against mostly American air forces.
There have also been a number of incidents similar to today's shooting down of a Russian aircraft: see here for details of the US-USSR shooting down a variety of each others' aircraft between 1950 and 1970
Note that this link is only describing USA-USSR interactions (not post-USSR Russia nor other non-US NATO members), in a specific timeframe (1950-1970), and only counts incidents where one aircraft was shot down by another aircraft (so doesn't include aircraft shot down from the ground, or any other land or sea incidents).
I don't know off the top of my head if there are other incidents of any type. I would assume there were border tensions around the Berlin Wall (/Inner German Border) and other borders which probably let to some level of clashes at various stages, and there have almost certainly been incidents between submarines that we never hear full details of.