Yes, but probably not in the way you were thinking. You'd need to use Google's Gmail for Work program and set things up on that side, but that would mean that you'd only be able to use a no-reply sending address from the Free Hosting web server, since sending through an external SMTP connection isn't allowed (due to heavy abuse for spam and phishing).
If you need something more seamless/less messy, you'd need to use paid hosting. In fact, if you want something reliable enough to build a business on, you should really be using paid hosting anyway. A service level agreement (SLA) and paid support staff beats the heck out of volunteer support staff who may or may not notice that you've reported a problem because somebody's hijacked your support forum thread. Hosting is cheap, especially when you compare it to the cost of losing the whole business because you tried to save the less than $0.25/day a reliable setup would cost. Free Hosting is great for stuff that doesn't really matter all that much in the big picture, but if your company really is more than a hobby with narcissistic personality disorder, web hosting isn't the place to cut all the corners; there's too much to lose and not enough to save.