Nope, the error message is 100% accurate. InDesign CS was written to run on the old G3/G4/G5 PowerPC-processor Macs. It would work on older versions of OS X on Intel processors via Rosetta (which is Power-PC emulation on Intel processor), but 10.7 no longer has Rosetta, so you can't run InDesign CS on 10.7.
You could set up a virtual install of an older version of OS X (10.6 would be the last compatible OS) in something like VirtualBox or VMWare Fusion or some other virtualization tool. Alternately, if you live in a place that has a market for used Macs, I'd tell you to go out an buy an old Mac. That is probably the cheapest, easiest way to continue using your old software. I have an old mirror-door G4 that I start up two or three times a year to work in InDesign CS or PageMaker or other old software.
In design CS won't open in OSX 10.7. Is there a quick fix vise new purchase? I get a error message that says (you cant open InDesign CS because PowerPC applications are no longer supported).
This is a great question, at least for me. I've often wondered about it. I run Windows on my Mac with Parallels.
I never thought about what you've asked, so I asked Google: "can parallels or vmware run earlier mac os x versions on a later mac os x" without quotes.
Here's the first link that I saw that seemed to be on point:
A quick scan notes two approaches; get a new external drive, or use virtual software like Parallels or VMware. It mentions a no-no about running Snow Leopard (OS X 10.6) in a virtualizer, but it goes on to say that 10.6 Server is OK. The article recommends a link to another article that compares VMware vs. Parallels.
The real "trick" might be finding a legal copy of OS X 10.6 SERVER, or 10.5 or earlier.