In terms of galaxies, no. The Andromeda Galaxy is approaching us, but Galaxies not in our local group are generally receding from us.
In terms of stars in our own galaxy, blue shift is quite common. In fact, that's how scientists figured out in which direction our sun is orbiting and rougly
how fast the period of rotation around the center of the galaxy is.
Stars closer to the center of the galaxy and ahead of us in their orbits generally show a red shift- they're rotating faster than the sun and receding from us.
Stars closer to the center of the galay and behind us in their orbits generally show a blue shift- they're rotating faster than the sun and getting closer to us.
View attachment 206770
Conversely, stars in the opposite quadrants, farther from us are the opposite. Depending on your orientation, stars in Quadrants I and III may show a red shift,
stars in Quadrants II and IV a blue shift.