@paok1970,
It's definitely something people say in the United States. I'm sure it's said in the UK too.
Examples:
1. It'll be easier to get the job if you have the right connections.
2. How on earth did she get to where she is? She had the right connections.
3. It's important to have the right connections if you want to thrive in this company.
Sorry they're all job/power related. That's what I usually associate this phrase with.