I'm pretty certain that you can because Gmail has places to import and export their e-mails.
Hi,
I'm not sure what you mean exactly - do you want to log in your Gmail using your Facebook account? If so, there's an easy way you can do that: You can just simply connect your Facebook account to your Google one following some simple steps. All you need to do is just work with the settings of your Google account, that you can find when you log into Gmail. Here is the video that explains it all:
Also, for more information, you can follow the official Gmail page on Facebook that has all sorts of tips and are 100% reliable.
I hope this was what you were looking for!