In short, no it is not possible for LinkedIn to use Google's cookies to suggest new contacts for you.
Cookies for most sites just contains an id value that really only means something to that particular site's server. You can view what values are contained in a particular cookie in most browsers. In Chrome this is located under the Developer Tools (Ctrl+Shift+I) on the Resources Tab. In addition most browsers only allow a site to access cookies that were set by that site.
It is possible that LinkedIn used your Google contact list to suggest new people, but this occurs only if you explicitly entered your Gmail username and password into their contact importer.
Most social networks suggest people you might know based upon the number of connections you have in common, the interests you share and the organizations you are both members of. Each social network has its own algorithms for suggesting common contacts, which they keep private. However there are many public academic studies researching how social networks are formed, basically it boils down to the fact that social networks typically are made up of small groups which have a lot of connections in common. In that group there are a few people that act as connectors to other small groups.