You could try stone effects spray paint. Make sure it’s rated for exterior applications. :)
Judy: here is a good video for your project!
I second using an outdoor spray paint that is gray and has texture. If you're filling it with water for the birds though, I'm not sure if spray painting the inner bowl will be safe for them to drink. I have a bird bath in my rose garden and I drilled some small holes in the bottom for drainage, filled it with dirt and turned it into a fairy garden with some succulents and annuals. Maybe you could do something like that.
How about gluing some pebbles in the portion where the birds will be drinking? (They don't like slippery surfaces). Then fill in around the pebbles with a dark grout to match the stone paint you used on the rest of the bird bath. You won't have to worry about toxins from the paint leeching into the water.