Here is a really simple way to search through someone's Twitter feed / stream with Google, to find a specific word or phrase.
To search a Twitter stream or feed simply type " site:twitter.com/USERNAME " into the normal Google search box - without the quotation marks and replacing 'username' with the Twitter address of the person or organisation your're interested in.
After this type the word or phrase you're searching for.
For example, to find what I have been saying about the climate on my Twitter account you'd type into a Google search: site:twitter.com/christiandunn climate
For most feeds Google seems to have around a two week delay in indexing tweets, but other than that it works pretty well.