Sorry, but I don't think so! For one thing, Katy Perry's real name is Katheryn Hudson, and she chose the name Katy Perry to avoid confusion with the actor Kate Hudson. 'Perry' is Katy's mother's maiden name.
Katy and Rick - Miles Apart
Rick Perry was born in Texas in 1950 and has two children. Katy was born in California and has a sister and a brother, so the details don't really tie up. Perhaps the only thing in common is as a staunch Republican, and Christian, Rick Perry might not approve of hits like 'I Kissed a Girl and I Liked It,' and maybe Katy's parents felt the same way! Katy is also a gay rights activists and sang at concerts in support of Barack Obama's re-election campaign, so it's just as well they're not related.
Here's Katy accepting an award for her work for gay rights: