This is probably a very stupid question but I need to hear it from others just to make sure and ease my worries. Will Google think these are different links now? I can't see how it would but everything amazes me with the bots. If, for some reason, it would think these are new links, we can't make the change.
Please, any advice here would be greatly appreciated before our team starts to recode.
The SE's will see them as the same, so do not worry. Though we don't always do it ourselves, it is generally considered a SEO best practice. The more complex a site is, the more important it is to use absolute links, which prevent the SE's from getting "confused" and making mistakes.