To answer this question generically, yes, a link back to your home page will pass some value, as long the page and link is crawled and indexed, the link is not nofollowed, and the site/page is not in some sort of penalization state.
That said, the amount of value is dependent on literally hundreds of factors and that would be a major discussion.
What I think might help is for me to differentiate two (of many) different types of links, the link you describe is an internal link. An internal link would be one from one page of a particular site to a page within that same site. That link gets valued differently than an external inbound link. An external inbound link would be one from another site to your site.
So, does it count, most likely. Is it counting in the same sense of how you are thinking about ‘backlinks’……it’s hard to say.
I would recommend that you do some additional research on link development related to the difference between inbound externals, link value pass-through (often referred to as link juice), internal links, link assessment based on position, site authority, domain authority, page authority, etc. and it may give you a better sense of how complex this question is. It will put the effort required to build a comprehensive link development strategy in perspective and allow you to put together an overall ‘off-page’ organic search strategy for your site.