It depends on the case-sensitivity of your webserver/OS.
If your website is hosted on a case-sensitive OS (like Linux) then Google will need to re-index those pages for your site to work. The old (mixed case) url would result in a 404 not found. You would need to set up 301 redirects to prevent your site from breaking and inform the search engines of the change. The same as if your page had moved to an entirely different URL.
If your OS is not case-sensitive then both URLs will work. The URLs in Google index will return a 200 (OK) as before, so as far as Google is concerned, nothing has changed. If you change all your links (as presumably you would) then maybe Google will pick up the change in URL over time, but maybe not - but neither would it really matter.
Should you change to lowercase URLs when your server is case-insensitive?
URLs are commonly all lowercase in order to avoid confusion and user error in a case-sensitive environment. If your server is not case-sensitive then it doesn't really matter from a users perspective. For consistency you might consider changing to lowercase, but it's really your decision. If there is ever a chance in the future that your site will be moved to a case-sensitive environment then changing to all lowercase now would perhaps be a good move.
It should be noted that, according to the spec, URLs should be compared in a case-sensitive manner (regardless of what your server is currently doing). Various analytics software (Google Analytics included) see URLs in a case-sensitive manner (by default). So, whilst your system might not be case-sensitive, other systems you interface with might be.